between systems used to make predictions and systems whose behavior one tries to predict.
For if meaningful general predictions are to be possible, it must at some level be the case that the system making the predictions be able to outrun the system it is trying to predict. But for this to happen the system making the predictions must be able to perform more sophisticated computations than the system it is trying to predict.
In traditional science there has never seemed to be much problem with this. For it has normally been implicitly assumed that with our powers of mathematics and general thinking the computations we use to make predictions must be almost infinitely more sophisticated than those that occur in most systems in nature and elsewhere whose behavior we try to predict.
But the remarkable assertion that the Principle of Computational Equivalence makes is that this assumption is not correct, and that in fact almost any system whose behavior is not obviously simple performs computations that are in the end exactly equivalent in their sophistication.
So what this means is that systems one uses to make predictions cannot be expected to do computations that are any more sophisticated than the computations that occur in all sorts of systems whose behavior we might try to predict. And from this it follows that for many systems no systematic prediction can be done, so that there is no general way to shortcut their process of evolution, and as a result their behavior must be considered computationally irreducible.
If the behavior of a system is obviously simple—and is say either repetitive or nested—then it will always be computationally reducible. But it follows from the Principle of Computational Equivalence that in practically all other cases it will be computationally irreducible.
And this, I believe, is the fundamental reason that traditional theoretical science has never managed to get far in studying most types of systems whose behavior is not ultimately quite simple.
For the point is that at an underlying level this kind of science has always tried to rely on computational reducibility. And for example its whole idea of using mathematical formulas to describe behavior makes sense only when the behavior is computationally reducible.