Mathematics is a human creation. A few extremely simple mathematical ideas are there in infants. The rest has been created by human mathematicians, using basic (and largely unconscious) cognitive mechanisms discovered in the cognitive and brain sciences, such as frames, conceptual metaphors, image-schemas, and neural bindings. The idea of infinite things (infinite numbers, points at infinity, infinite sets, infinite decimals, etc.) arises from a single conceptual metaphor used in many branches of mathematics. Concepts like imaginary numbers, logarithms, trigonometric functions, etc. also arise via frames, metaphorical concepts, and neural bindings. Mathematical ideas are like other ideas, but there is a set of constraints that make them mathematical ideas and not just ideas.

This understanding of mathematics is scientific in nature, coming from the brain and cognitive sciences. It utterly undermines the romantic idea that mathematics is just out there in the world, or in some Platonic universe. It also undermines the usual accounts of the “foundations of mathematics” — Platonism, formalism, intuitionism, and logicism. All of these conflicting “foundations” are also remarkable inventions of human mathematicians with human brains.

The implications for the teaching of mathematics at all levels are revolutionary. From this perspective, mathematics becomes understandable to ordinary human beings.