Computational Platonism

Abstract

We offer a view of mathematics as an experimental science where axioms play the role of foundational theories like general relativity and quantum mechanics in physics. Under this view, axioms are provisional and inferred from experience with the experiental substrate of mathematics which we locate within computation rather than encoding intuitive and absolute truths. This offers a reframing of Godel's theorem, placing its impact sharply upon the incompleteness rather than the potentially contradictory nature of any computational set of axioms. The essay originated in an attempt to make precise the nature of mathematics in order to estimate how AI might impact it. This exploration is continued in the paired essay "The mechanical creation of mathematical concepts."