Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Other algebraic systems

Of algebraic systems studied in traditional mathematics the vast majority are special cases of either groups, rings or fields. Probably the most common other examples are those based on lattice theory. Standard axioms for lattice theory are (\[Wedge] is usually called meet, and \[Vee] join)

{Wedge[a, b]==Wedge[b, a], Vee[a, b]==Vee[b, a], Wedge[Wedge[a, b], c]==Wedge[a, Wedge[b, c]], Vee[Vee[a, b], c]==Vee[a, Vee[b, c]], Wedge[a, Vee[a, b]]==a, Vee[a, Wedge[a, b]]==a}

Boolean algebra (basic logic) is a special case of lattice theory, as is the theory of partially ordered sets (of which the causal networks in Chapter 9 are an example). The shortest single axiom currently known for lattice theory has LeafCount 79 and involves 7 variables. But I suspect that in fact a LeafCount less than about 20 is enough.

(See also page 1171.)

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