From: hal@finney.org
Date: Sat Mar 16 2002 - 16:05:27 MST
Eliezer writes:
> Hi, guys! I just figured out a way to - without ontotechnology - insert a
> short natural-language message into Pi, starting at (for the sake of the
> discussion) around the 1000th digit, and continuing for one or two extended
> sentences, maybe 30 to 100 letters total. I guess it just goes to show that
> you should never leap to conclusions about what post-Singularity entities
> can and can't do!
The only way I had thought of to do something like this would be to
control which mathematical constants turn out to be found interesting
by the likely inhabitants of a universe.
Our universe is 3 dimensional, gravity pulls things into balls with 2D
surfaces, plus we write and draw on 2D surfaces. This calls our attention
to the mathematics of plane geometry and we soon learn about Pi. It is
am important constant for us.
But in the abstract mathematical sense, Pi is not all that simple. You
need to create axioms to represent all of Euclidean geometry, plus
arithmetic, to even be able to write a formula for Pi as we know it.
(Actually you don't really need geometry, you can do it with infinite
series. Pi/4 = 1 - 1/3 + 1/5 - 1/7 + .... But that's still quite a
few axioms just to define the numbers and arithmetic.)
You can think of an axiom system as a string of characters, or even
more simply as a string of bits which acts as a computer program which
can generate theorems or verify proofs. The length of this program
then represents the complexity of the mathematical system.
It is plausible that the minimum mathematics needed to define Pi is
similar in size to what Eliezer describes above, a few hundreds or
thousands of bits. Given this fact, there are a similar number of other
constants which might be defined using the same amount of information.
Each of those constants would be "simple" from the perspective of the
particular mathematical system defined by those axioms. But with that
many different constants to try, it should be possible to find one which
has the desired message embedded in it, yet is fundamentally no more
mathematically complex than Pi.
All that remains is to create a universe which somehow embodies the
mathematical axioms which lead to the constant that has the message
you want to convey. That seems to be the hard part, because presumably
only a tiny fraction of all theoretical universes are even capable of
evolving intelligent life. The universe creator needs to satisfy two
constraints at once, that the universe evolve intelligence and that the
required mathematics is fundamental and important. But perhaps for a
powerful enough universe creator this is a solvable problem.
In such a universe, some other mathematical constant would have a
similarly fundamental role as Pi does for us. To the inhabitants of
that universe it would seem impossible that their constant could have
been manipulated by anyone. It has an inherent mathematical simplicity
and elegance which would make inserting a message impossible, just as
we consider to be true for Pi. But in fact the message is sent, not by
changing a constant, but by controlling which constant the inhabitants
discover.
Hal
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