Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Generators and relations [and axiom systems]

In the axiom systems of page 773, a single variable can stand for any element—much like a Mathematica pattern object such as x_. In studying specific instances of objects like groups one often represents elements as products of constants or generators, and then for example specifies the group by giving relations between these products. In traditional mathematical notation such relations normally look just like ordinary axioms, but in fact the variables that appear in them are now assumed to be literal objects—like x in Mathematica—that are generically taken to be unequal. (Compare page 1159.)

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]