Brent Meeker writes:
> Does Wolfram have a critereon for discriminating between "random" and
> "sturctured", or is it "just how it looks?"
I am only 1/4 way through the book, but at this point he seems to mostly
be basing his criteria on how the output looks. I understand from book
reviews that at some point he will bring in the notion of computational
systems that have the power of universal computers, and show that some
simple systems are able to do this. These are all inherently of class 4
(structured), I believe.
The first 6 (of 12) chapters, which are as far as I have read, are
essentially an observationally-oriented discussion of the behavior of
simple computing systems. Wolfram reminds me of an 18th century scientist
exploring a new part of the natural world, say an early microscopist
or astronomer. He is just trying to get the basic classifications into
place, observing similarities among different systems. But instead of
studying plants and animals, he is studying computational systems.
In our terms, this portion of the book can be thought of as a naturalist's
beginning explorations of the multiverse, the collection of all universes
produced by computer programs. His main result so far seems to be that
it doesn't matter what is the basis for your computational engine: CAs,
Turing machines, numeric recurrence relations, partial differential
equations; all produce the same basic classes of patterns. He examines
10-20 different ways of implementing simple computational systems and
shows that they all look much the same. The main difference is that
some produce structure in as many as ~10% of programs, while for others
it is only perhaps one in a million.
I see some connections between Wolfram's explorations and Greg Chaitin's
(creator of Algorithmic Information Theory) results. For years, Chaitin
has emphasized the need to approach mathematics as an experimental
science, but no one has taken him very seriously. Wolfram's work suggests
to me an approach to putting this program into action. For example,
here is a quote from Chaitin I ran into last night, from a recent essay,
A case in point is elementary number theory, where there are some very
difficult questions. Consider the prime numbers. Individual prime
numbers behave in a very unpredictable way, if you're interested
in their detailed structure. It's true that there are statistical
patterns. There's a thing called the prime number theorem that predicts
fairly accurately the overall average distribution of the primes. But
as for the detailed distribution of individual prime numbers, that
looks pretty random.
So I began to think that maybe the inherent randomness in mathematics
provided a deeper reason for all this incompleteness.
Wolfram has a chapter on patterns in the integers, and several of his
examples of randomness are based on the distribution of primes. Is the
randomness of this distribution caused by the same effects which Wolfram
has identified, which make so many simple computational systems produce
similarly random distributions? If we understand that randomness is a
frequent consequence of simple systems, then it is not surprising that
primes are random in the details of their distribution.
Another connection between Chaitin and Wolfram has to do with the
underpinnings of the multiverse. From Chaitin and AIT we know that any
universal computer can emulate any other using a fixed size program.
This means that universes agree on their relative measures to within a
constant factor (whose value depends on which system is emulating which
other system). Therefore, in the limit of large programs, where this
constant factor becomes insignificant, universes have the same relative
measures independent of what the details are of the computational
substrate which runs the multiverse. (I interpret this to mean that we
can never find out what kind of computer it is that the multiverse is
based on, or equivalently, that it doesn't matter.)
This seems to somewhat contradict Wolfram's experiments, which show that
different computational systems have rather different (a few orders of
magnitude) probabilities of showing complex behavior. Presumably the
reason is that Wolfram is only dealing with simple programs. If it were
possible to extend his results to larger programs, we might predict
based on AIT that all computational systems would converge to equal
probabilities for the different classes of behavior.
I understand that Wolfram introduces some ideas about computational
equivalence in the last chapter of the book, but I don't think he goes
in exactly this direction.