Algorithmic theories of everything (TOEs) are limited to universe
histories describable by finite computer algorithms. The histories
themselves may be infinite though, computable by forever running programs.
To predict a future we need some conditional probability distribution
on possible futures, given the past. Algorithmic TOEs are limited
to distributions computable in the limit, that is, there must be some
(possibly forever running) finite computer program that approximates
with arbitrary precision the probability of any possible future.
What about NONalgorithmic TOEs? For instance, once one assumes that
the cardinality of possible futures equals the cardinality of the real
numbers one has a NONalgorithmic assumption.
I postulate that the restricted, algorithmic TOEs are preferable over
nonalgorithmic TOEs, for two reasons: they are simpler, yet there is
no evidence that they are too simple. They are simpler not just in some
vague, model-dependent, quantitative way but even in a fundamental and
qualitative way, because they are fully describable by finitely many
bits of information, while NONalgorithmic TOEs are not. We can write
a book that completely describes an algorithmic TOE. We cannot write a
book that completely describes a nonalgorithmic TOE.
Algorithmic TOEs vs nonalgorithmic TOEs - it's really like describable
things vs nondescribable things.
I join those who claim that things one cannot describe do not exist.
For instance, most real numbers do not exist. Huh? Isn't there a
well-known set of axioms that uniquely characterizes the real numbers?
No, there is not. The Loewenheim-Skolem Theorem implies that any first
order theory with an uncountable model such as the real numbers also
has a countable model. No existing proof concerning any property of
the real numbers really depends on the "continuum of real numbers",
whatever that may be. Our vague ideas of the continuum are just that:
vague ideas without formal anchor.
Some might think there is an algorithmic way of going beyond
algorithmic TOEs, by writing a never halting program that outputs
all beginnings of all possible universe histories in a continuum
of histories --- let us call this an "ensemble approach":
http://www.idsia.ch/~juergen/everything/html.html
But while there is a clever ensemble approach that needs only countably
many time steps to output ALL infinite histories that are INDIVIDUALLY
computable within countable time, and to output their FINITE complete
descriptions, there is no ensemble approach that outputs ALL histories of
a continuum:
http://rapa.idsia.ch/~juergen/toesv2/node27.html The ensemble
approach is firmly grounded in the realm of algorithmic TOEs, and cannot
go beyond those. An ensemble approach can output all incompressible finite
strings, all with finite descriptions, but it cannot output anything that
is not computable in the limit. In particular, the ensemble approach
cannot provide us with things such as infinite random strings, which
simply do not exist!
The algorithmic constraint represents a strong, plausible, beautiful,
satisfying bias towards describability. What we cannot even describe
we must ignore anyway. The algorithmic constraint by itself is already
sufficient to show that only histories with finite descriptions can have
nonvanishing probability, and that only histories with short descriptions
can have large probability:
http://rapa.idsia.ch/~juergen/toesv2/ More
is not necessary to demonstrate that weird universes with flying rabbits
etc. are generally unlikely. Why assume more than necessary?
Provocative (?), actually straightforward conclusion:
henceforth we should ignore nonalgorithmic TOEs.
Received on Thu Feb 08 2001 - 09:04:37 PST