Showing Text From Page | View Full page with images

that they ultimately tend to be equivalent in their computational sophistication—and thus show all sorts of similar phenomena.

And what we will see in this section is while some of these phenomena correspond to known features of mathematics—such as Gödel's Theorem—many have never successfully been recognized.

But just what basic processes are involved in mathematics?

Ever since antiquity mathematics has almost defined itself as being concerned with finding theorems and giving their proofs. And in any particular branch of mathematics a proof consists of a sequence of steps ultimately based on axioms like those of the previous two pages [773, 774].

The picture below gives a simple example of how this works in basic logic. At the top right are axioms specifying certain fundamental equivalences between logic expressions. A proof of the equivalence Nand[p, q] == Nand[q, p] between logic expressions is then formed by applying these axioms in the particular sequence shown.