“This is an essay within a new field of study – the cognitive science of mathematics. […] You might think that this enterprise would leave mathematics as it exists alone and simply add to it an account of the conceptual nature of mathematical understanding. You could not be more wrong. Studying the nature of mathematical ideas changes what we understand mathematics to be and it even changes the understanding of particular mathematical results.”
"One of the properties of commonplace conceptual metaphors is that they preserve forms of inference by preserving image-schema structure. [...] One of the reasons why the inference structures of mathematical proofs is stable is that the inference structures of commonplace metaphors is stable. That feature of metaphors is one of the reasons why theorums, once proved, stay proved."