Alastair Malcolm wrote:
>I have been looking through the archives on this list and cannot find any
>clear arguments against the following challenge to Tegmark's hypothesis (
>'Is the theory of everything merely the ultimate ensemble theory?'):
>
>If our world can be equated to a mathematical structure which is related
>by the physical laws that are apparent to us, then it can also be a far
>more complex mathematical structure which explicitly specifies the
>universe as it has evolved during our lifetimes (for example a phase space
>specification of all the particle positions/momenta for a universe coming
>into existence say in 1850, and happening to obey
>classical/quantum-mechanical laws as required to convince us that the
>universe does follow simple laws).
>
>Now there would seem to be far more different mathematical structures
>where this type of scenario occurs (but with sufficient deviation from
>'normality' such that we would notice - the odd white rabbit scuttling
>across a ceiling, for instance, to reuse an earlier example), than there
>is of the relatively simple mathematical structure(s) that science implies
>underlies our phenomenal world. So statistically we should be in one of
>these 'contrived' universes. (This also happens to be the essence of what
>I take to be John Leslie's position against Tegmark's - and David Lewis's
>- hypotheses, though he has expressed it differently.)
>
>Perhaps I could play the role of devil's advocate for the moment, and say
>that if no counter-argument can be found to this challenge, then Tegmark's
>scheme must be rejected out of hand, because we see no levitating rabbits.
>(I would also just like to say that I am specifically addressing Tegmark's
>scheme described in the paper mentioned above, and not any TOE entirely
>confined to a MWI of QM (which, as a 1b category hypothesis would be
>vulnerable to the criticism mentioned in his paper), nor Shmidhuber's
>'Great Programmer' hypothesis (which would seem to need to address a
>'turtles all the way down' accusation).)
>
>Does anyone happen to have an idea about how to respond to this challenge
>to Tegmark's hypothesis?
I believe that this challenge MUST be met with any "everything exists"
hypothesis.
But the challenge CAN be met only if the "everything" is defined in a
sufficiently effective way.
By taking "all mathematical structures" Tegmark will be unable to define
and calculate an effective probability if only because the collection of
all
mathematical structures cannot be defined mathematicaly.
This, if I remember correctly, is also Wei Dai's motivation for
computationalism (look at the beginning of this discussion list).
If physicists take Tegmark seriously they will fall in the infinitely
delicate problem bearing on the very foundation of mathematics.
Here the computationalist hypothesis, and especially Church's thesis is
welcome because, as Goedel eventually realised, it introduces in math a
kind
of absolute epistemological concept (computability, by opposition to
provability). They are also nice invariant theorem (like the compiler
theorem cited by Schmidhuber, which make "your" challenge tractable in
this frame.
I agree with George Levy that any restriction on "everything" is
arbitrary.
But, and that is the root of Church's thesis power, the computationnalist
apparent restriction is not really a restriction. It is only a restriction
if you believe that "uncountability" or "unprovability" are absolute
notions.
This is not the case (cf Skolem paradox, ...).
Could you tell us if you agree that Schmidhuber "great programmer"
approach
should also met the flying rabbit challenge ?
And could you explain more precisely the 'turtles all the way down'
accusation. (I guess you confuse internal and external "time").
Bruno
Bruno MARCHAL Phone : +32 (0)2 6502711
Universite Libre Fax : +32 (0)2 6502715
de Bruxelles Prive : +32 (0)2 3439666
Avenue F.D. Roosevelt, 50 IRIDIA, CP 194/6
B-1050 BRUSSELS Email : marchal.domain.name.hidden
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Received on Mon Jul 05 1999 - 02:23:19 PDT