From Euclidean Geometry to Euclidean Spaces
YAP Von Bing
National University of Singapore
Singapore
Published in The Mathematician Educator, 2025, Vol 6, No. 2, pp 47-71.
Abstract
Kronecker famously said “God made the integers, all else is the work of man.” Here is a sketch of how Euclidean spaces are motivated by Euclidean geometry, which is viewed explicitly as a science. The unusual approach highlights the abstractness and hence utility of Euclidean spaces, and offers an intuitive interpretation of the Cauchy-Schwarz inequality.