Until 1800, mathematics was divided into two great branches: geometry and arithmetic. Both were commonly regarded as the more obviously secure repositories of human knowledge. At this stage, geometry could be suitably defined as "the science which investigates the properties and relations of magnitudes in space, as lines, surfaces, and solids" (Oxford English Dictionary). However, with the enormous enrichment of mathematics in the nineteenth century, the scope of geometry was greatly expanded and diversified, its content disrupted, and its epistemic standing called into question.

The word "geometry" comes from a Greek word that literally means measurement of the earth and was originally applied to the art of land surveying. But around 500 BCE or even earlier, the spatial properties and relations that had been codified by land surveyors in Mesopotamia and Egypt became in Greece the starting-point of inquiries of a more abstract sort that soon took leave of their...