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In most kinds of mathematics there are all sorts of additional details, particularly about how to determine which parts of one or more previous expressions actually get used at each step in a proof. But much as in our study of systems in nature, one can try to capture the essential features of what can happen by using a simple idealized model.

And so for example one can imagine representing a step in a proof just by a string of simple elements such as black and white squares. And one can then consider the axioms of a system as defining possible transformations from one sequence of these elements to another—just like the rules in the multiway systems we discussed in Chapter 5.

The pictures below show how proofs of theorems work with this setup. Each theorem defines a connection between strings, and proving the theorem consists in finding a series of transformations—each associated with an axiom—that lead from one string to another.

But just as in the multiway systems in Chapter 5 one can also consider an explicit process of evolution, in which one starts from a