[FOM] Principle of Computational Equivalence

[Todd Rowland]

>All that now remains are the geographical/biological problems, to sort out
>where this holds, and where it doesn't.  An important task, for science.
>But again, all of it that seems to be contained in the PCE is the fact
>that "UTMs exist".  I can't help feeling that I'm missing something vital.

Coming from my point of view, the PCE's important point is that 
universality is common.  Beyond a simple existence claim, it says that if 
you see something in nature that is complicated, then that thing is likely 
to be universal.  To make use of the PCE as a guiding principle, one has 
to develop an intuition for judging complex behavior -- in particular, 
recognizing repetition and nesting as not being complex.

The PCE also covers abstract systems.  If you search a space of 
computational rules starting from the simplest cases, then you will 
quickly find examples of simple rules that are universal.  For instance, 
looking for the simplest universal Turing machine is best done by 
searching.

For other people's opinions about the PCE, I suggest asking such questions 
on the NKS forum:
http://forum.wolframscience.com

Todd Rowland
Wolfram Research

P.S. Many of the projects at the NKS Summer School involve the PCE one way 
or another:
http://www.wolframscience.com/summerschool