**The Principle of Computational Equivalence**

We were pleased to read that a three color machine is the simplest possible
universal Turing machine and congratulate
the winner who proved it.

http://blog.wolfram.com/2007/10/the_prize_is_won_the_simplest.html

At this occasion we turn again to NKS Chapter 12 and read: “In essence,
therefore, the Principle of Computational Equivalence introduces a **new law
of nature** to the effect that no system can ever carry out explicit
computations that are more sophisticated than those carried out by systems like
cellular automata and Turing machines.” Reading this we wonder whether this
“law of nature” applies also to life?

What we perceive around us is **change**. An ongoing and continuous change
to which even Kant’s “the thing in itself” does not apply. There is not a
thing out there, only change. Fortunately we are equipped with two faculties
for handling this change: A memory to store it and a mind to interpret it. For
interpreting change, mind applies innate and acquire memories. While the
change there is essentially continuous our mind freezes and makes it discrete.

Wolfram regards **change as a process**, which may be regarded as a computation,
assuring us that “all processes, whether they are produced by human effort or
occur spontaneously in nature, can be viewed as computations” and “Computational
Equivalence applies to essentially any process of any kind, either natural or
artificial.” Therefore the “Principle of Computational Equivalence can be
viewed in part as a new law of nature.”

Wolfram thus presumes that this “law of nature” governs also change. A Platonic
idea which we mortals cannot grasp since it is obscured by change. So far so
good, yet what about continuity? Cellular automata and most of the other
computational systems that are discussed in his book are discrete**. **Does
it imply that the change that surround us is also discrete?

Wolfram: “It is my strong suspicion that at a fundamental level absolutely
every aspect of our universe will in the end turn out to be discrete. And if
this is so, then it immediately implies that there cannot ever ultimately be
any form of continuity in our universe that violates the Principle of
Computational Equivalence.”

Wolfram: “In a sense the most basic defining characteristic of continuous
systems is that they operate on arbitrary continuous numbers. But just to
represent every such number in general requires something like an infinite
sequence of digits. And so this implies that continuous systems must always in
effect be able to operate on infinite sequences.”

What Wolfram prefers to ignore is that **mathematics, be it discrete or
continuous is a way to describe change**, and cannot fully reproduce the
continuity of the change that surrounds us. Thus Computational Equivalence is
far from being a new law of nature**. **It is a creation of Wolfram’s mind
which is busy interpreting the ongoing continuous change. It may be regarded
as an important law of mathematics, a valuable tool for modeling phenomena of
nature, but not a law of nature.

*Vive la petite difference!*