But the discussion of universality in the previous chapter already suggests that this is not the case. For it implies that at least across the kinds of systems that we considered in that chapter there is in fact an upper limit on the sophistication of computations that can be done.
For as we discussed, once one has a universal system such a system can emulate any of the kinds of systems that we considered—even ones whose construction is more complicated than its own. So this means that whatever kinds of computations can be done by the universal system, none of the other systems will ever be able to do computations that have any higher level of sophistication.
And as a result it has often seemed reasonable to define what one means by a computation as being precisely something that can be done by a universal system of the kind we discussed in the previous chapter.
But despite this, at an abstract level one can always imagine having systems that do computations beyond what any of the cellular automata, Turing machines or other types of systems in the previous chapter can do. For as soon as one identifies any such class of computations, one can imagine setting up a system which includes an infinite table of their results.
But even though one can perfectly well imagine such a system, the Principle of Computational Equivalence makes the assertion that no such system could ever in fact be constructed in our actual universe.
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.
So what might make one think that this is true? One important piece of evidence is the success of the various models of natural systems that I have discussed in this book based on systems like cellular automata. But despite these successes, one might still imagine that other systems could exist in nature that are based, say, on continuous mathematics, and which would allow computations more sophisticated than those in systems like cellular automata to be done.
Needless to say, I do not believe that this is the case, and in fact if one could find a truly fundamental theory of physics along the lines I