>
> In Tegmark's paper,
> in section 2G, he makes a crucial point that the fewer axioms
> you use to define your mathematical structure, the larger is
> the ensemble. This provides a concrete justification for the
> principle of Occam's Razor. Similarly to the argument given
> above, we would expect to find ourselves in worlds with fairly
> few laws of physics, since those admit the most SAS's. You
> can always add any bizarre behavior to the structure by adding
> ad hoc axioms, but worlds in which that is the case
> have a smaller measure than those that do not.
>
> This line of reasoning also explains why, in a general sense,
> we find that our universe behaves sensibly from moment to moment.
> Many philosophers have pondered the question of why everything
> doesn't disintegrate into chaos in the next instant. What holds
> the world together such that things persist and our memories
> match our external reality? The answer is that the structure(s)
> we are in obey physical laws, not because they were cast by
> fiat from some omnipotent being, but simply because the structures
> that do obey physical laws are more numerous than those that do
> not, and hence we are likely to find ourselves in those.
>
I would take issue with this last statement. It seems that the above
argument would imply that chaotic, unlawful universes should be more
numerous than those obeying laws. The usual justification given for us
finding ourselves in a universe with physical laws, is that such a
universe is a minimum requirement for concious beings (or SASes, to
use Tegmark's terminology) to exist. Unfortunately, because we don't
understand the nature of conciousness enough, we cannot predict what
the most general mathematical structure is to contain SASes.
One approach I suggested earlier is to suggest that perhaps the
structure underlying QM (ie a Hilbert space over the complex numbers)
is the most general such structure, and work backwards, trying to find
the reasons behind the properties.
A Hilbert space is a Vector Space (ie it elements have a linearity
property with respect to some field), and it has the additional
property of having an inner product <x,y>.
QM has an additional property of the universe being indexed by time
"t". Schroedingers equation can be written as e^{iHt}, where H is a
Hermitian operator. This basically follows from some reasonable
assumption about probability (i.e. the probabilities of all
possibilities must remain equal to 1 through time). I think it
reasonable to assume that conciousness requires a time variable.
The inner product is required to form "projections", which are the
basis of observations: P(x is observed when A is measured)=<x,Ay>
where y is the state of the universe. Something of this nature is
required for conciousness.
If one has a vector space, then the most general field possible is the
set of complex numbers, so that explains why C is used.
The problem is - why linearity? Is it related to Bayesian laws of
probability - ie that probabilities of independent events are additive
( P(AuB) = P(a)+P(B) & P(AnB) = P(A)P(B) )? Is it necessary to have
independent events for conciousness? Is it possible to get Bayesian
laws with some generalisation of linearity?
NB. QM is in fact inconsistent with our other big 20C theory of
Physics, ie Relativity. One of the biggest problems is that in
Relativity, there is no well defined concept of "now" - the locus of
contemporary events depends on one's frame of reference. One of the
most interesting attempts to reconcile these theories replaces them
with motion on a fractal manifold, and time has a 2D character. It is
only in the macroscopic limit that time has a one-dimensional
nature. Perhaps the Anthropic Principle requires the universe only to
be approximately described by QM to some level of accuracy. (We
needn't worry about solutions that blow up over periods much longer
than the current age of the universe).
Enough ranting for now.
>
>
> --
> Chris Maloney
> http://www.chrismaloney.com
>
> "Knowledge is good"
> -- Emil Faber
>
>
>
----------------------------------------------------------------------------
Dr. Russell Standish Director
High Performance Computing Support Unit,
University of NSW Phone 9385 6967
Sydney 2052 Fax 9385 7123
Australia R.Standish.domain.name.hidden
Room 2075, Red Centre
http://parallel.hpc.unsw.edu.au/rks
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Received on Sun Jun 06 1999 - 17:43:43 PDT