CMP Review 2023-05-25

CMP Review 2023-05-25

May 25, 2023

In the 17th century, René Descartes introduced Cartesian coordinates. David Nelson describes the impact: “Their discovery allowed the application of algebraic methods to geometry and the study of hitherto unknown curves.”

Prior to Descartes, one might be forgiven for thinking of geometry and algebra as two fundamentally separate disciplines, as distinct as, say, grammar and arithmetic. But Descartes enabled us to see that x² + y² = r² is as accurate and valid a definition of a circle as the set of points equidistant from a center.

In the 20th century, graphing calculators were introduced to the world. They allow us in a moment to view the spatial representation hidden as it were in a string of symbols and numbers. Some of these curves are novel, some beautiful, all interesting.

Of course we learn how to plot these curves by hand. Of course we learn all about conic sections. But sometimes it’s nice to enjoy the fruit of the centuries and witness in a heartbeat the image behind some equation we are manipulating. It is a reminder that no matter how deep we are in algebra, geometry is never far away.

But such a perspective requires a certain view of math. A view that math is really a domain of wonder. A belief that all these mathematical signs are really symbols of living ideas, not mere ciphers, no more fit for humans than for machines.