Introduction

This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book describes the psychological basis for mathematics at every level. It explains some of the innate tendencies humans have, from birth, that support the construction of numbers, and returns to the construction of numbers. The book provides an illustration of that program with quadrilaterals and extends it as a psychological foundation for all of mathematics. It demonstrates the psychological basis for one of the most famous theorems in mathematics: the Pythagorean theorem. The book also provides a rigorous method for symbolizing mathematical objects while offloading their demands on working memory. It focuses on covariation in the context of trigonometric functions, concluding with a surprising connection to prime numbers, constructible polygons, and Pingala’s triangle. The book introduces a framework for understanding variables and covariation as a coordination of our own mental actions.