Pythagoras’ theory of everything: quantum mechanics from the natural numbers

The natural numbers play a key role in axiomatic set theory and hence in the foundations of mathematics. The natural numbers also have a major role in quantum mechanics. The idea is explored that von Neumann’s construction (vNC) of the natural numbers, within the framework of Zermelo–Fraenkel (ZF) axiomatic set theory, can serve as a blueprint for deriving some key basic equations of quantum mechanics. This approach obviates any need for quantizing classical mechanics and provides further support for the view that quantum mechanics is perfectly applicable to fields beyond those in physics for which it was originally intended. This article is part of the theme issue ‘Quantum theory and topology in models of decision making (Part 1)’.