Is the Mandelbrot Set real?

John K Clark (johnkc@well.com)
Fri, 18 Jul 1997 21:36:49 -0700 (PDT)

-----BEGIN PGP SIGNED MESSAGE-----

On Mon, 14 Jul 1997 Hara Ra <harara@shamanics.com> Wrote:

>with the Mandeobrot Fractal, we may explore admittedly inexact
>representations to the degree that our computational resources allow.
>However, this does not say the Mandelbrot Set exists as a complete
>object somewhere, only that we can explore some aspects of it in a
>finite manner.

Is the Mandelbrot Set real? Well, there are some points that we know for sure
are in The Mandelbrot Set and some points we know for sure are not. However,
there are other points that we'll never be able to determine to be in the set
or not, and to make things even worse, we can't discriminate between those
impossible points and those who's status is just difficult to determine.
The bottom line is that we can't know everything about The Mandelbrot Set,
and can't even define the limits of our ignorance, but is that really any
different from any other object? Also, does OUR ability to have knowledge of
an object determine if it really exists or not? Actually, if some (not all)
of the interpretations of Quantum Mechanics are correct then the answer to
that last question is yes.

This brings up a related question, I've heard it said that The Mandelbrot Set
is the most complex object in mathematics, but is it? The shape is
fantastically complex, as well as beautiful, and it would take a computer an
infinite amount of time to render it, but the program itself is amazingly
simple, only a few lines long, hell I wrote a Mandelbrot program when I was a
kid. How could a dumb kid who didn't know his ass from a hole in the ground
write a program that could, given enough time, produced the most complex
object in Mathematics?

Complexity is not as clear cut an idea as it would seem, even the formal
definitions contain subjective elements, and they don't seem to correspond
very well with our intuitive understanding of the word. By some standards a
frog is more "complex" than a human, it certainly contains a lot more DNA.

Some say AIC is a measure of complexity, but there are problems.
The Algorithmic Information Content (AIC) of a message string can be
found by measuring the length of the smallest computer program that can
produce the string, and that's just another way of asking how much the string
can be compressed. The AIC is largest for random strings because the
smallest program that can produce a truly random string is the string itself,
it's completely incompressible. Accordingly, the gibberish a monkey produces
by banging on a typewriter contains far more information than a Shakespeare
sonnet of the same length. This is almost the exact opposite of what we
usually mean when we say something is complex. Another rather serious problem
with AIC is that we can never find it, we can only find an upper bound.
Greg Chaitin proved a few years ago that in general you can't be certain that
there is not a program shorter than any you know about that will generate the
string. This also means that although most strings are random and thus
incompressible, you can't prove that any particular string is random.

Charles Bennett of IBM thinks that in addition to AIC we need another measure
of complexity he calls "Depth", it's a measure of how long a program must run
to produce its result, not how big the program is. The trillionth's prime
number or the Mandelbrot set can be generated with very short programs, but
they must run for a VERY long time, so they would have very little AIC but
enormous depth.

Depth helps but it doesn't solve everything, the works of a monkey are still
more complex than Shakespeare. I think part of the problem is that all
information is not of equal value, and it's difficult to get a handle on that
mathematically. I also think it would be a mistake to equate complexity with
virtue, in fact all things being equal, the opposite is true. A simple
program is better than a complex one that does the same thing. A simple
ecosystem such as a temperate forest of oaks and conifers is more resilient
than a very complex one like a tropical rain forest.

>Shades of Wittgenstein (I hope I spelled it right).

You did, but you got Mandelbrot wrong.

John K Clark johnkc@well.com

-----BEGIN PGP SIGNATURE-----
Version: 2.6.i

iQCzAgUBM9BBsn03wfSpid95AQEKiQTsCQ8MnuJCik4/WY92sqfarmaxXS7qUJnM
st2ja7nMlYlQ4unFxGy6MqupQMemKrtZErOB3RA1FVnV5g54jPOa2F6E0W7aE09P
q1X+ec/S26sZDuWmGnXHCeDww6rJpNmgfEZANz0snBRGpdqyeRL4lokmz6VrW4iI
y3Qdl27H+1t5t6htyVEeoqkuv5tabcYNi11SQYt4vWLEMBgG+Do=
=WB6M
-----END PGP SIGNATURE-----