We investigate the Folium of Descartes, viewing it both as a cubic curve in the plane, and as a surface in three dimensional space. This is an extended exercise in algebraic calculus.We parametrize it, calculate Taylor polynumbers and expansions and tangents, and interpret them both as tangent planes of tangent lines, or as tangent quadrics or tangent conics. We finish with a pleasant duality satisfied by the tangent conics.This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics.
Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.