Indeed, in an extreme case it might even be possible to do the analog of what has been done, say, in the computation of symbolic integrals, and to set up some kind of uniform procedure for finding a proof of essentially any short theorem.

And in general whenever there are enough repeated elements within a single proof or between different proofs this indicates the presence of computational reducibility. Yet while this means that there is in effect less new information in each theorem that is proved, it turns out that in most areas of mathematics these theorems are usually the ones that are considered interesting.

The presence of universality implies that there must at some level be computational irreducibility—and thus that there must be theorems that cannot be reached by any short procedure. But the point is that mathematics has tended to ignore these, and instead to concentrate just on what are in effect limited patches of computational reducibility in the network of all possible theorems.

Yet in a sense this is no different from what has happened, say, in physics, where the phenomena that have traditionally been studied are mostly just those ones that show enough computational reducibility to allow analysis by traditional methods of theoretical physics.

But whereas in physics one has only to look at the natural world to see that other more complex phenomena exist, the usual approaches to mathematics provide almost no hint of anything analogous.

Yet with the new approach based on explicit experimentation used in this book it now becomes quite clear that phenomena such as computational irreducibility occur in abstract mathematical systems.

And indeed the Principle of Computational Equivalence implies that such phenomena should be close at hand in almost every direction: it is merely that—despite its reputation for generality—mathematics has in the past implicitly tended to define itself to avoid them.

So what this means is that in the future, when the ideas and methods of this book have successfully been absorbed, the field of mathematics as it exists today will come to be seen as a small and surprisingly uncharacteristic sample of what is actually possible.