Robin Hanson, <[EMAIL PROTECTED]>, writes:
> 1) There's a certain elegant simplicity to the claim "all possible universes 
> exist", at least if "possible" is interpreted as broadly as possible, i.e.,
> "not logically inconsistent".  But if you start substituting other meanings 
> for "possible", I think the elegance quickly disappears.  You'd then still
> have to explain why other non-logically-inconsistent universes don't exist,
> and you'd have a bunch more weird universes to explain why we don't easily 
> observe them.   

Is logical consistency well defined?  For example, one possible universe is
Conway's Life game.  What would be the equivalent of logical consistency
in that universe?

Is the idea that we find a mapping between symbolic logic and structures
in the universe, and that a universe for which such a mapping does not
exist is not logically consistent?

Maybe logical consistency is not a necessary prerequisite for a universe
to exist.  One of the points of the Schmidhuber paper was that physical
laws would not necessarily be followed in the future, because there
would exist universes like ours in the past but unlawful in the future.
Colloquially we might say that a universe which suddenly abandons
following natural laws is not logically consistent.  But we should not
require this kind of consistency if we can avoid it (i.e. come up with
some reason why the universe does appear to be lawful).

> I find the many worlds theory of quantum mechanics elegant, but not because
> it can be thought of as equally the above claim with a certain odd quantum
> definition of "possible".  I find it elegant because it seems simpler than
> the known alternatives which account for the empirical data.

I find them elegant for the same reason, though: MW avoids the need for
the constant creation of new information.  Without MW, each time the
wave function collapses we have to come up with new random bits to set
the physical values.  With MW, the program to run the universe is much
shorter and doesn't need to come up with all these values all the time.

Likewise if all universes exist we don't even have to hardwire the physical
laws of our universe, we can just iterate over all possible physical laws.
Again, the program to run the universe(s) is much smaller.

> 3) My basic problem with the "all possible universes exist" claim is that
> I find it hard to figure out whether my actions have any consequences.  
> If all possible universes exist, then for any me in one universe choosing 
> one action, there is another me in another universe choosing another action.
> In a global sense I can't choose actions anymore.  All possible actions get
> chosen.  

Of course this would not go to the truth or falsity of the claim, simply
a hard problem if it turns out to be true.

I think a necessary ingredient is that we have to assume that there is a
probability distribution over the universes.  Even if all universes exist,
our chances of experiencing any conceivable universes is not equal.
We pretty much are forced to believe this anyway by the fact that the
universe keeps on being lawful.

Given that, even if you do make all choices, the probability distribution
over them is not even.  Some choices are more probable than others.
Without trying to get too far into the difficulties of free will in
what is now a deterministic multiverse, you could consider that your
deliberations and decisions influence this probability distribution.
Hence you have control over the fraction of the multiverse in which you
take one action vs another.

Hal

FFrom [EMAIL PROTECTED]  Tue Jan 20 15:06:23 1998
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Wei Dai, <[EMAIL PROTECTED]>, writes:
> What is the set of all possible universes? Max Tegmark says its the set of
> all mathematical structures, and Juergen Schmidhuber says its the set of
> all Turing machines, but neither gives much justification. I tend to agree
> with Schmidhuber, if only because Tegmark's definition does not seem to
> lead to an effective theory. For example, what does a uniform distribution
> on all mathematical structures mean? However it would be nice to have some
> stronger justifications for assuming that only computable universes exist.

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.

> If we agree that all Turing machines exist, it's still not clear how they
> should be interpreted as physical universes. Schmidhuber suggests that the
> output of a Turing machine should be interpreted as the evolution of a
> universe. However this is problematic because it does not lead to an easy
> way to think about the complexities and probabilities of structures within
> a universe. For example, consider a simple Turing machine that enumerates
> the natural numbers. The output of this TM includes every possible
> configuration of every other universe. What is the probability that I'm
> living in this TM? I don't see a straightforward way to answer this
> question under Schmidhuber's interpretation. The problem is that even
> though this universe is very simple, it would take a lot of extra
> information to find anything in it, so the simplicity of the universe as a
> whole is deceptive.

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.

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.

Consider a more plausible candidate for the states of our universe.
Each short substring represents the physical presence or absence of an
elementary particle at a particular location of spacetime.  The time
slices are at the Planck scale and are with respect to some arbitrary
but realistic reference frame.

Suppose this interpretation does lead to a universe which looks just
like ours.  Now, there would be a mapping from the simple counting
string to this one, but the mapping would be as large as the program
which created this realistic universe.  So that shouldn't count.

The mapping from strings to universe states should be simple and obvious,
and if it's not, change the TM program until you get output strings which
do have a simple and obvious mapping.

> As an alternative, I suggest that each Turing machine should be thought of
> as taking the coodinates of a region of a universe as input and producing
> the content of that region as output. Here region should be broadly
> interpreted. It not only refers to volumes of space-time, but may also for
> example specify a branch of a quantum superposition.

I think you have a mapping problem here as well.  With enough creativity
in defining my coordinate system I could map our universe to your counting
TM.  Basically I think your definition will be equivalent to the other
one in terms of program size.

Hal

rom [EMAIL PROTECTED]  Tue Jan 20 15:06:24 1998
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In Schmidhuber's model, the Turing machine creates strings composed of
zeros, ones, and commas.  We then interpret the commas as separators and
get a series of numbers.  We then further interpret the series of numbers
as universe states indexed by time.

(This interpretation has some obvious limitations; all universes must
have discrete time steps, and the amount of information at each time
index must be finite.  Probably a more elaborate mapping could address
these, although it's not clear that a universe with general relativity
and quantum mechanics fits very well into this scheme.)

Given this kind of interpretation, the TM is superfluous at this level.
We could simply say, "all strings exist", and interpret certain strings
as corresponding to certain universes.

The TM comes into play when we want to get a probability distribution.
This is done based on the length of the shortest program which emits a
given string; universes with shorter programs are assumed to be more
probable, if we are so inclined.

An equivalent formulation would be to go back to the strings and say that
the probability of a given string is based on its algorithmic complexity.

A question here is whether the strings actually have to be written down or
otherwise instantiated in some universe in order to allow our perceptions
to exist.  Likewise does the TM in Schmidhuber's model actually have to
run somewhere?

Tegmark takes a Platonic view, the idea being that a sufficiently complex
mathematical system inherently leads to what feels like a real universe.
Nobody actually has to write down the equation, or run the simulation.
>From the inside, we can't tell whether that is being done or not.  This
viewpoint could apply to the string or TM models as well.

A Platonic model make it harder to see why a probability distribution
should exist.  But even one where the TM has to exist doesn't really
explain what the a priori probablities should be very well, IMO.

Hal

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