Notes

Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations


Reducing axiom [system] details

Traditional axiom systems have many details not seen in the basic structure of multiway systems. But in most cases these details can be avoided—and in the end the universality of multiway systems implies that they can always be made to emulate any axiom system.

Traditional axiom systems tend to be based on operator systems (see page 801) involving general expressions, not just strings. But any expression can always be written as a string using something like Mathematica FullForm. (See also page 1169.) Traditional axiom systems also involve symbolic variables, not just literal string elements. But by using methods like those for combinators on page 1121 explicit mention of variables can always be eliminated.


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