Are thoughts REALLY Random?

[Jordan B Pollack]

**Don't Forward this one**

Richard just reminded me that I recently read Penrose's unfortunately
named book, making a nice Dancing Woolly (thats Wu Li) appear like
Surly Lion (thats Searlie).  Which, by the way, reminds me of a topic
to bring up for discussion, now that Turing has finally been grounded,
at least symbolically.

You see, Penrose went on and on about his new-age religious experience
of accessing the Platonic Universe of Absolutely True Mathematical
Ideas (He obviously didn't finish Jaynes), and how complex and beautiful
mathematical structures, such as Mandelbrot's set, complex numbers,
and his own tilings, are in "there" to be discovered, rather than
invented, and would continue to exist whether we found them or not.
This universe seems like a pretty big space to me; if only I could dip
into it to create publications!

So, consider when a computation (or a mathematician) transfers a
collection of "information" from the infinite (but ghostly) Platonic
Universe into our own finite (but corporeal) universe, by generating a
fractal picture or writing down a brand-new theorem. How can this
bunch of bits be quantified?

Apologizing for the brutal reduction of other's lifework, I know about
counting binary distinctions (Shannon), I know about counting
instructions (Kolmogorov), I know that only a truly random string is
really complex (Chaitin), I know that machines can be ordered
lexicographically (Godel), and I even know that the computable
languages form a strict hierarchy (Chomsky).

However, since programs of the same size, when executed, can yield
structures with vastly different APPARENT randomness and bit-counts,
some useful measure of the "Platonic density" seems to be missing
from the complexity menu.  I find it difficult to believe that the
Mandelbrot set is only as complex as the program to "access" it!

Can somebody please help me make sense out of this? 

Jordan Pollack                            Assistant Professor
CIS Dept/OSU                              Laboratory for AI Research
2036 Neil Ave                             Email: pollack at cis.ohio-state.edu
Columbus, OH 43210                        Fax/Phone: (614) 292-4890