Newton, the towering scientific figure of the 17th century, discovered a lovely method for finding approximate solutions to equations, involving iterated constructions of tangent lines and their intersections. We describe this method in general and then apply it to the simplest and most familiar example; the standard quadratic polynomial x^2. To calculate tangent lines we use the algebraic calculus, and focus on approximating sqrt(2). This gives, remarkably, the same algorithm as the ancient Babylonians had for approximating sqrt(2), and so another alternative to the Vedic procedure.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.