Hi Bruno,

   "Once more unto the breach, dear friends, once more"!

-----Original Message----- 
From: Bruno Marchal 
Sent: Sunday, April 03, 2011 1:03 PM 
To: everything-list@googlegroups.com 
Subject: Re: Causality = 1p Continuity? 


>> On 03 Apr 2011, at 05:15, stephenk wrote:

snip
>>
>> [SPK] That logical structures alone are insufficient to model our
>> existence.

> Correct. But arithmetical structure are enough (or please mention a  
> flaw in UDA).

[SPK] 

    I wish to be doubly sure that I am not arguing against a straw man, 
therefore I will be quoting from and commenting on:

 
http://groups.google.com/group/everything-list/browse_thread/thread/4c995dee307def3b

“COMP is the hypothesis that there is a level such that I 
survive a digital functional substitution of my generalised body/brain 
made at that level, + Church Thesis (CT: digital = turing) + Arithmetical 
Platonism (AR: the belief that arithmetical propositions obeys 
classical logic, and this independently of my own cognitive ability). 

To sum up: COMP = \exists n SURV-SUBST(n)    + CT   + AR “


“b) CU: there is a Concrete Universe, whatever it is. This is need 
   for the decor. 

c) CUD: there is a Concrete running of a UD in the concrete universe. 


d) 3-locality: computations are locally implementable in the 
   concrete universe. That is it is possible to separate two 
   implementations of two computations in such a way that the result 
   of one of these computations will not interfere with the result 
   of the other one. Computations can be independent. 
   More generally the result of a computation is independant of 
   any event occuring a long way (out of the light cone) from that 
   computation.”

...

“12) A Universal Dovetailer exists. (Extraordinary consequence of 
Church thesis and Arithmetical Realism). The UD simulates all 
possible digital devices in a quasi-parallel manner). 

(Adding a line in the code of any UD, and you get a quasi- 
computation of its Chaitin \Omega number). 

13) So let us assume CU and CUD, that is let us assume explicitly 
there is a concrete universe and a concrete running of a UD in it. 
This need a sort of steady state universe or an infinitely expanding 
universe to run the complete infinite UD. 
Suppose you let a pen falls. You want predict what will happen. 
Let us suppose your brain is in state S at the beginning of the 
experiment. The concrete UD will go to that state infinitely often 
and compute all sort of computational continuations. This is 
equivalent to reconstitutions. It follows from 11 that your 
expectation are undetermined, and the domain of the indeterminism 
is given by the (infinite) set of reconstitutions. To predict, 
with COMP, what will happen you must take into account all 
possible histories going through the state S of your brain. 
And here clearly the NEURO hypothesis is not used. Even if your 
real brain state is the state of the actual concrete universe, 
with COMP that state will be generated (infinitely often) by the 
UD. Same reasoning if your brain state is the quantum state of 
the universe, so the reasoning works even if the brain is a 
non local quantum object (if that exists). So the physics is 
determined by the collection of your computational continuations 
relatively to your first person actual state.” 
14) If 'that' physics is different from the traditional empirical 
physics, then you refute COMP. But with COMP you will not refute 
COMP, isn't it? So with COMP you will derive the laws of physics, 
i.e. invariant and similarities in the 'average' continuations of 
yourself (defining the measure on the computationnal continuations). 


Exercice: why should we search a measure on the computational 
continuations and not just the computational states? Hint: with 
just the computational states only, COMP predicts white noise for 
all experiences. (ok Chris ?). With the continuations, a priori 
we must just hunt away the 'white rabbit' continuations. 
You can also show that Schmidhuber's 'universal prior' solution 
works only in the case the level of substitution 
is so low that my generalised brain is the entire multiverse. 
(see below). 


15) Once you explain why arithmetical machines are statistically right 
to believe in physical laws without any real universe, such a real 
universe is redundant. 
By Arithmetical Realism and OCCAM razor, there is no need 
to run the concrete UD, nor is there any need for a real concrete 
Universe. 
(Or you can use the movie graph argument to show that a first 
person is not able to distinguish real/virtual/and *Arithmetical* 
nature of his own implementations, and this eliminates OCCAM.)” 


   OK, my problem is that SURV-SUBST(n) requires that the UD actually run on 
some form of a CU as a CUD. You account for this by introducing CUD (CUD 
necessitates the existence of CU). The CU and CUD involve a measure of change 
that can be identified with “time” that is invariant under parameterizations 
(by the teleportation with delay argument), or equivalently there must exist a 
spatial distance interval, because there must be some arbitrary parameter to 
distinguish 3-localities from each other. If all 3-localities are exactly 
isomorphic in their content then by the law of indiscernibles they are all one 
and the same. All that one would have, maybe, is endomorphic maps from the 
3-local to the 3-local, but even those would require the existence of a 
concrete structure that is their dual.
    
    But this invariance does not eliminate the fact that the UD must be run to 
be said to generate the digital simulations that are equivalent (under 10 –13 
of UDA) to 1p and their continuations. This parameterization invariant notion 
of time (or space) is necessary for any expression/implementation of AR. AR 
must be expressible as some belief in each 1p (modulo coherent and soundness): 
for AR to exist then it is necessary that a 1p believe that AR exists and the 
statement “AR exists” is true. If the belief that AR exists cannot be expressed 
by a CUD then AR cannot be said to exist since it would be impossible to 
express the statement “AR exists”. Diagonalizations require some form of CU 
support or else they all collapse into Nothing.

    You then argue that AR + OCCAM allows you to eliminate CUD and thus CU. But 
there is a problem with this! AR necessitates CUD to be distinct from Nothing 
(per R. Standish’s definition and argument 
http://www.amazon.com/Theory-Nothing-Russell-Standish/dp/1921019638/)! Just 
because “Real” and “Virtual” are 1p indistinguishable (which is an isomorphism) 
does not necessitate that Arithmetic representations of the 1p = 1p (which is 
an identity). To do this is to violate the Representation theorem because an 
isomorphism is not an identity, it is a mapping between two distinct entities. 
   
1) SURV-SUBST(n) implies the existence of 1p
2) SURV-SUBST(n) necessitates CUD.
3) 1p necessitates CUD.
4) if CUD does not exist, then neither does SURV-SUBST(n) and thus 1p does not 
exist.
5) if 1p does not exist, then AR cannot be expressed since AR/1p = Nothing.
6) If AR cannot express on any 1p, then AR cannot exist.
7) Thus if CUD does not exist, then AR does not exist.
QED.

You are assuming that AR can exist and be expressive without any support or 
supervenience. How is this so? Consider how AR must exist as distinct from 
Nothing otherwise AR is equivalent to Nothing and have no properties or 
orderings or valuations or distinguishing features or properties at all. We see 
in the Representation theorem that “every abstract structure with certain 
properties is isomorphic to a concrete structure (such as a transformation 
group on some set.).” http://en.wikipedia.org/wiki/Representation_theorem 
    For AR to exist as distinct from Nothing then there must exist a concrete 
structure, a CU, that it is isomorphic to and yet is distinct from by any 1p. 
This is especially important in light of the fact that the CU that is necessary 
for the UD requires a parameter invariant notion of interval and this 
requirement cannot be achieved if AR = Nothing. This kind of flaw flows, in my 
humble opinion, from the mistake of assuming that because we can map the 
sequencing of events in a 1p history to the positive Reals then the sequence of 
events of all 1p = the positive reals. This reasoning fails because of the 
requirements of general covariance that is an empirically verified fact of our 
experienciable reality. 

    General covariance demands that for all of the representations of the 
symmetry groups of the CU there exists a smooth diffeomorphism between them for 
all 1p; all observers must see the same form of physical laws otherwise there 
is a preferred frame of reference. A preferred frame of reference is equivalent 
(via your “real is indistinguishable from virtual” argument!) to a special 1p 
that can act as a computational oracle to decide whether or not any given 
generic 1p contains self-contradictory information, white rabbits, cul-de-sacs, 
etc. I believe that the “measure” that you keep referencing is just another 
form of this oracle. If that oracle exists then P=NP! See: 
http://en.wikipedia.org/wiki/P_versus_NP_problem

   In conclusion: Unless one has something to be mindful of there is no need to 
have a mind at all. A mind at least must have a concrete implementation of 
itself to be able to exist for some other mind. A mind that does not exist for 
any other Mind has not means to define itself as distinct from Nothing.
    

>> We need the physical world to be the interface between our
>> separate minds,

> Eventually with comp, the physical world is recovered by defining it  
> as an interface between our different minds, or as the gluing dreams  
> processes. We need a physical world. No doubt on this. The point is  
> that we don't need a primary physical world.

[SPK] 
    I agree 100%. We do not need a “primary physical world”. But by my argument 
above we do need some non-primary form of CU to run the UD so that AR can be 
expressed. Unless there exists a CU there cannot be a AR since AR is isomorphic 
to some CU per the representation theorem. It is necessary for both Abstract 
and Concrete structures to exist as distinct from Nothing.


snip

  >> It is a young bipolar genius, of the kind "perishing (not
  >> publishing)". His only work are notes that he wrote to me with the
  >> solution of the first open math question in my thesis. I have put  
  >> them
  >> on my web pages. Here is the link:
  >>
  >> http://iridia.ulb.ac.be/~marchal/Vandenbussche/AxiomatisationZ.html
  >>
  >> The solution of the open problem is in the first three slides. It
  >> shows also that G and Z are bisimulable. The other slides comes from
  >> some questions I asked to him. It includes a pretty result showing
  >> that the sentences asserting their own Sigma_1 truth are false (a  
  >> sort
  >> of anti-Löbian phenomenon).
  >
  > [SPK] Could you elaborate on this bisimulation?

  The B of the logic Z can be define in G by Bp & Dt, and the D of Z, by  
  Bf v Dp (the D of Z is really the usual logican's notion of relative  
  consistency).
  Vandenbussche found that you can dually reverse that translation: the  
  B of G can be defined in Z by Bp v Df, and the D of G can be defined  
  in Z by Dp & Bt.
  Be careful to interpret the B and D in the right logic. I should  
  perhaps write this in the following less ambiguous (but less readable)  
  way:

  B_z A   ==   B_g A  &  D_g t
  D_z A   ==   D_g A  v  B_g f

  B_g A   ==   B_z A  v  D_z f
  D_g A   ==   D_z A  &  B_z t

  The two lines above are the usual definition of the Z box (the second  
  follows by duality on Bp & Dt)
  The two last lines are Vandenbussche inversion. It leads toward an  
  axiomatization of Z, Z1, Z* and Z1*.

  So despite their very different semantics, and "hypostasic role", G  
  and Z are variants of each other. The same for G1 and Z1, G1* and Z1*.

  Unfortunately there is no such transformation available for the logics  
  X.    (X, X1, X*, X1*)
  We conjecture that G and X are not bisimulable, nor probably S4Grz and  
  X.

[SPK]
    It is my conjecture that if a bisimulation does not exist for a given logic 
that can be expressed in a 1p then that logic (and its derivations) are purely 
“subjective”, e.g. given a pair of 1p it is not possible for 1p_1 to 
communicate anything about a non-bisimulatable logic to 1p_2 and vice versa. So 
by your comment G and X are purely subjective logics, aka pure solipsisms.

Onward!

Stephen

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