James, Doug, Bruno, Gilles:
thanks for your recent messages!
I apologize for being such a slow responder.
Bruno writes:
> Although most real numbers are not describable at all, we can easily =
> talk about the set of all real numbers. Now, the Great Programmer (my =
> Universal Dovetailer UD) not only dovetails on all computations but =
> dovetails on all computations with all real (complex, quaternionic, =
> ...) oracles. It does so by generating all finite initial segments of =
> all reals.
One should be careful here. Discussions of sets of noncomputable numbers
and proofs of certain theorems concerning their properties are computable.
Such computable aspects of R, however, should not distract from the
fundamental difference between N and R. Most of what's in R is completely
beyond computation.
Note that all your initial segments of reals simultaneously are initial
segments of some of the few computable reals (e.g., fractions). There are
countable computable reals but uncountable others. Hence most reals
are beyond dovetailing. From an information-oriented perspective, reals
tend to complicate things. Most reals don't fit the Great Programmer
religion, which is a simple one.
> At each state of the computation of the UD, there is =
> indeed no place for a real number, BUT, I show below that from the =
> first person perspective of an "observer", the set of all real number =
> does play an important role. In fact our prediction will depend on =
> the measure on all infinite computationnal histories. More below.
> What do you mean by "an observer inhabits U_k" ?
It means the string sequence E_k is being interpreted as representing
(among other things) some observer's life in U_k. The interpreter
(another observer, possibly a Great Programmer himself) may or may not
be part of E_k.
Abstract concepts such as recognition, interpretation, observation
(popular in our particular universe, not necessarily in others) are
vague. Apparently they refer to establishing a relationship between
an observed thing and the observer's previous knowledge. To repeat,
any non-trivial relationship of this kind implies that observation
and previous knowledge share mutual algorithmic information: there is
a comparatively short algorithm computing observation from knowledge.
Otherwise the observation will seem random or irregular (like noise)
from the observer's point of view.
> But I believe such an expression has no =
> (non-ambiguous) meaning once we take seriously the comp hypothesis. =
> An observer cannot so easily be said inhabiting a precise U_k. The =
> observer cannot be localized in a unique computation. He can only be =
> associate with an infinity of "sufficiently similar" computations =
Why? It all can happen in a particular computable world - no obvious
need to sum over "an infinity of sufficiently similar" computations."
Vague concepts such as "sufficiently similar" complicate things: making
them precise requires additional bits of information.
This reminds me of my brother (a theoretical physicist) who also used to
insist on summing over all computations. I guess it's the way physicists
get trained. But "summing over everything" basically seems covered by the
universal Solomonoff-Levin distribution (whose properties tend to remind
physicists of more familiar path integrals): most of the probability
of a given universe is in its simplest (shortest) program. So we may
basically ignore most of the many ways a particular universe can be
computed. Essentially we need to focus on just one program, not many.
> It is necessary to concentrate ourself on the following thought =
> experiments (PE, PE1, PE2, PE3, PE4, PE-omega).
I guess I have to admit you lost me here. You mention abstract high-level
concepts such as teletransport experiments, reconstitution, time-delays,
expectations, undeterminism, average robots. True, some of the Great
Programmer's universes may be interpretable (by appropriate computable
interpreters) as being inhabited by people who write messages about such
things. But I was not quite able to see how this affects the contents of
my little paper, whose main objective is to keep things simple, probably
because I am just a simple mind myself.
Gilles writes:
> So you are basically saying: the Universe is N. But why not R, P(R) or any
> more complicated set?
Because R is not the simplest thing compatible with what we know.
In absence of evidence that R is indispensable, Occam's razor leaves
us with N.
> It is not surprising that the people of the last XX century imagine the
> world based on a computation, just like Greeks have imagined it as a
> combination of four elements, or theologist have imagined it as driven by
> God (it confirms that consciouness can only produce ideas from the
> interaction with an outer world!)
Very true! People always try to come up with explanations that are
simple, given current knowledge. I often tried to ponder whether the
Great Programmer religion is just another example. I found comfort though
in the fact that it does encompass everything we will ever be able to
talk about (the noncomputable stuff is beyond description). So I like
to think it's the final example ... %^)
Juergen
www.idsia.ch
Received on Mon Mar 08 1999 - 01:23:27 PST
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