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.
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!