Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

One-way transformations [as axioms]

As formulated in the main text, axioms define two-way transformations. One can also set up axiom systems based on one-way transformations (as in multiway systems). For basic logic, examples of this were studied in the mid-1900s, and with the transformations thought of as rules of inference they were sometimes known as "axiomless formulations".

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