On Wed, 3 Jul 2002, Hal Finney wrote:
> 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.
..
> 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,
> http://www.cs.auckland.ac.nz/CDMTCS/chaitin/amsci.html:
>
> 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.
What's Wolfram's critereon for "randomness"? how it looks? It would seem
that he's using something different than Chaitin if we thinks a simple
system can produce random output. Chaitin *defines* random as an output
that can't be reproduced by a system simpler than the output itself.
> 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,
How's that? Isn't a constant factor is of the same significance for all
sizes? Are you thinking of an additive term instead of a factor (but that
doesn't apply to Turing emulations)?
Brent Meeker
"There is nothing more difficult to take in hand, more perilous to
conduct, or more uncertain in its success, than to take the lead in the
introduction of a new order of things."
--- Niccolo Machiavelli, in The Prince
Received on Wed Jul 03 2002 - 15:27:57 PDT