Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Character of mathematics

Since at least the early 1900s several major schools of thought have existed:

Formalism (e.g. David Hilbert): Mathematics studies formal rules that have no intrinsic meaning, but are relevant because of their applications or history.

Platonism (e.g. Kurt Gödel): Mathematics involves trying to discover the properties of a world of ideal mathematical forms, of which we in effect perceive only shadows.

Logicism (e.g. Gottlob Frege, Bertrand Russell): Mathematics is an elaborate application of logic, which is itself fundamental.

Intuitionism (e.g. Luitzen Brouwer): Mathematics is a precise way of handling ideas that are intuitive to the human mind.

The results in this book establish a new point of view somewhere between all of these.

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