Dear David,
 
    This is a very good post! I would like to point you to a proposal that Vaughan Pratt discusses in several of his papers found here:
 
 
 
    The basic idea goes like this:
 
A causes B if and only if the information of B implies the information of A.
 
or A => B iff A* <- B*
 
    This goes into a commutation diagram that can be chained:
 
... A ----> B -----> C ...
    |          |           |
... A* <== B*< == C* ...
 
    The idea is that physical events chain together only if their representations imply each other and the "arrows" of "causation" and "logical implication" go in opposite directions. In this way we can make sense of how the past is "so well behaved" while the future is wide open to possibilities that include pathologies. BTW, this idea is very much in line with Wheeler's "Surprise 20 Questions".
 
Kindest regards,
 
Stephen
 
----- Original Message -----
Sent: Monday, November 03, 2003 10:45 PM
Subject: Is the universe computable?

In the words of Tegmark, let’s assume that the physical world is completely mathematical; and everything that exists mathematically exists physically.

 

I have been thinking along these lines since my days at university - where it occurred to me that any alternative is mystical.  However,  the problem remains to explain induction - ie the predictability of the universe.   Why is it that the laws of physics can be depended on when looking into the future, if we are merely a “mathematical construction” - like a simulation running on a computer.  It seems to me that in the ensemble of all possible computer simulations (with no limits on the complexity of the “laws”) the ones that remain well behaved after any given time step in the simulation have measure zero. 

 

Given the “source code” for the simulation of our universe, it would seem to be possible to add some extra instructions that test for a certain condition to be met in order to tamper with the simulation.   It would seem likely that there will exist simulations that match our own up to a certain point in time, but then diverge.  Eg it is possible for a simulation to have a rule that an object will suddenly manifest itself at a particular time and place.   The simulated conscious beings in such a universe would be surprised to find that induction fails at the moment the simulation diverges.

 

In other words, at each time step in a simulation the state vector can take different paths according to slightly different software programs with special cases that only trigger at that moment in time.  It seems that a universe will continually split into vast numbers of child universes, in a manner reminiscent of the MWI.  However there is a crucial difference – most of these spin off universes will have bizarre things happen.  It is difficult to see how a computable system can be tamper proof.  How can a past which has been well behaved prevent strange things from happening in the future?

 

In the thread “a possible paradox”, there was talk about a vanishingly small number of “magical” universes where strange things happen.  However, it seems to me that the bigger risk is that a “normal” universe like ours will be the atypical in the ensemble!

 

A possible argument is to invoke the anthropic principle – and suggest that our universe is predictable in order for SAS’s to evolve and perceive that predictability.  However,  that predictability only needs to be a trick – played on the inhabitants for long enough to develop intelligence.  There is no reason why the trick needs to continue to be played.

 

I suggest that the requirement of a tamper-proof physics is an extremely powerful principle.   For example, we deduce that SAS’s only exist in mathematical systems that aren’t computable.   In particular our Universe is not computable. 

 

- which is what Penrose has been saying.

 

I have assumed that non-computability coincides with being tamper-proof but this is far from clear.  For example, it is conceivable that the Universe is a Turing machine running an infinite computation (cf Tipler’s Omega point), and “awareness” only emerges in the totality of this infinite computation.  Perhaps our awareness is a manifestation of advanced waves sent backwards in time from the Omega Point!

 

I think it’s important to distinguish between an underlying mathematical system, and the formal system that tries to describe it.   I think this is a crucial distinction.   For example, the real number system can be defined uniquely by a finite set of axioms.   Uniqueness is (formally) provable - in the sense that it can be shown that an isomorphism exists between all systems that satisfy the axioms.    However the real numbers are uncountably infinite - and therefore are very poorly understood using formal mathematics - which is limited to only a countably infinite set of statements about them.   So formal mathematics should be regarded as an imperfect and coarse tool which only gives us limited understanding of a complicated beast!  This is after all what Godel’s incompleteness theorem tells us. 

 

It is not surprising that a computer will never exhibit awareness - because it is merely using the techniques of formal mathematics, and not tapping into the “good stuff”.

 

- David

 

 

 

 

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