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From: Russell Standish <R.Standish.domain.name.hidden>

Date: Mon, 7 Jun 1999 10:44:38 +1000 (EST)

*>
*

*> 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

----------------------------------------------------------------------------

Received on Sun Jun 06 1999 - 17:43:43 PDT

Date: Mon, 7 Jun 1999 10:44:38 +1000 (EST)

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.

----------------------------------------------------------------------------

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

----------------------------------------------------------------------------

Received on Sun Jun 06 1999 - 17:43:43 PDT

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