Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Rings [and axioms]

The axioms given are for commutative rings. With \[CirclePlus] being + and \[CircleTimes] being × the integers are an example. Several examples of rings arose in the 1800s in number theory and algebraic geometry. The study of rings as general algebraic structures became popular in the 1920s. (Note that from the axioms of ring theory one can only expect to prove results that hold for any ring; to get most results in number theory, for example, one needs to use the axioms of arithmetic, which are intended to be specific to ordinary integers.) For non-commutative rings the last axiom given is replaced by (a\[CirclePlus]b)\[CircleTimes]c == ( a\[CircleTimes]c)\[CirclePlus](b\[CircleTimes]c). Non-commutative rings already studied in the 1800s include quaternions and square matrices.

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