Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Symbolic systems [and operator systems]

By introducing constants (0-argument operators) and interpreting \[SmallCircle] as function application one can turn any symbolic system such as e[x][y]->x[x[y]] from page 103 into an algebraic system such as (e · a) · b == a · (a · b). Doing this for the combinator system from page 711 yields the so-called combinatory algebra {((s · a) · b) · c == (a · c) · (b · c), (k · a) · b == a}.

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