ABSTRACT
This chapter introduces the Erlangen program. First proposed by mathematician Felix Klein in the late 1800s, the Erlangen program used group theory to classify different geometries (e.g., Euclidean geometry and projective geometry). The chapter introduces the formal definition of a group and uses symmetries of quadrilaterals to exemplify them while illustrating the Erlangen program. Then, building on Chapters 1 and 2, the chapter extends the Erlangen program to include both number and geometry. Specifically, numbers, shapes, and mathematical objects in general, arise from a coordination of mental actions that are reversible and composable—two key properties of groups. Thus, the Erlangen program might be extended to explain the psychological foundation for all of mathematics.
