On Tue, Jan 20, 1998 at 02:46:54PM -0800, Hal Finney wrote:
> I think there would be a mapping between mathematical structures and TM
> programs.  Starting with a mathematical structure, described as a formal
> system, we could have a TM start writing out theorems in that system.
> In the other direction, we could create a mathematical description of
> any given TM.

But there are some mathematical strctures that can't be described as a
formal system. For example the set of all true statements about integers,
or a universe that contains an oracle for the halting problem. One can
even argue that most mathematical structures can not be described as
formal systems, since the set of all mathematical structures is not
countable. (Isn't every real number a mathematical structure? How about
every function on the real numbers?)

> Using Schmidhuber's mapping, the universe states would be 0, 1, 10, 11,
> 100, 101, 110, 111, ....  I don't see any simple mapping which would make
> this look at all like the universe we live in.  For example, one of the
> characteristics of this kind of counting is that there is a lot of change
> at the right hand side and very little at the left, and we don't really
> see that kind of behavior in our universe, which is somewhat homogeneous.

This argument doesn't work because we can't directly observe the state of
the universe and its evolution, so we can't say that it is not changing
only at the right hand side. If we accept Schmidhuber's interpretation,
there is no way to rule out the possibility that we're living in the
counting universe because everything eventually appears in it during its
evolution, including us. If we were living in the counting universe, of
course all of our memories and perceptions would have no basis in reality. 
So my problem with Schmidhuber's interpretation is even ignoring the
mapping problem, I don't see how it would assign a low probability to us
being in the counting universe. Perhaps he would comment if I'm
misunderstanding him.

> Now, maybe there could be some more complex mapping, where bits in this
> counting string get mapped sparsely throughout the universe in some
> complicated way.  This mapping problem is one which AI philosophers face
> as well: there is a mapping between the thermal motions of the atoms of
> a stone and the neuron firing patterns of a brain.  I think the solution
> is that in some sense the mapping has to be simple, or else the mapping
> is doing all the work.

I think the solution to the mapping problem is to assume no mapping. A
Turing machine is a universe, and its output given some region as input IS
the content of that region, not merely the encoding of the content of that
region. Otherwise, as you point out, you could hide arbitrary amount of
complexity in the encoding function.

The interpretation I gave earlier already implicitly assumes no mapping,
so I don't think it suffers from the mapping problem.

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