George Lakoff wrote a great book on this called "Where Mathematics Comes From." He is critical of how mathematics is currently taught from "first principles" a

George Lakoff wrote a great book on this called "Where Mathematics Comes From." He is critical of how mathematics is currently taught from "first principles" and proofs rather than from a more intuitive body/metaphor perspective. He believes proofs are in many ways academic gatekeeping. His final chapter explains Euler's identity using only the intuitive metaphors he built up in previous chapters.