Saibal Mitra wrote:

> I think the source of the problem is equation 1 of Juergen's paper. This
> equation supposedly gives the probability that I am in a particular
> universe, but it ignores that multiple copies of me might exist in
> one universe. Let's consider a simple example. The prior probability
> of universe i (i>0) is denoted as P(i), and i copies of me exist in
> universe i. In this case, Juergen computes the propability that if you
> pick a universe at random, sampled with the prior P, you pick universe
> i. This probability is, of course, P(i). Therefore Juergen never has
> to identify how many times I exist in a particular universe, and can
> ignore what consciousness actually is.  Surely an open universe where an
> infinite number of copies of me exist is infinitely more likely than a
> closed universe where I don't have any copies, assuming that the priors
> are of the same order?

To respond, let me repeat the context of eq. 1 [In which universe am I?]  
Let h(y) represent a property of any possibly infinite bitstring y, say,
h(y)=1 if y represents the history of a universe inhabited by yourself
and h(y)=0 otherwise.  According to the weak anthropic principle, the
conditional probability of finding yourself in a universe compatible with
your existence equals 1.  But there may be many y's satisfying h(y)=1.
What is the probability that y=x, where x is a particular universe
satisfying h(x)=1? According to Bayes,
P(x=y | h(y)=1) = 
(P(h(y)=1 | x=y) P(x = y)) / (sum_{z:h(z)=1}  P(z)) 
propto P(x),
where P(A | B) denotes the probability of A, given knowledge of B, and
the denominator is just a normalizing constant.  So the probability of
finding yourself in universe x is essentially determined by P(x), the
prior probability of x.

Universes without a single copy of yourself are ruled out by the weak
anthropic principle.  But the others indeed suggest the question: what can
we say about the distribution on the copies within a given universe U (maybe
including those living in virtual realities running on various computers in U)?
I believe this is the issue you raise - please correct me if I am wrong!
(Did you really mean to write "i copies in universe i?")

Intuitively, some copies might be more likely than others. But what
exactly does that mean? If the copies were identical in the sense no
outsider could distinguish them, then the concept of multiple copies
wouldn't make sense - there simply would not be any multiple copies. So
there must be detectable differences between copies, such as those 
embodied by their different environments.

So my answer would be: as soon as you have a method for identifying and
separating various observer copies within a universe U, each distinguishable
copy_i is different in the sense that it lives in a different universe
U_i, just like you and me can be viewed as living in different universes
because your inputs from the environment are not identical to mine.

In general, the pair (U_i, copy_i) conveys more information than U by
itself (information is needed to separate them).  The appropriate domain
of universes x (to use the paper's notation) would be the set of all possible
pairs of the form (separate universe, separate observer).

Equation 1 above is perfectly applicable to this domain.


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