Explore the foundations of calculus through an algebraic approach to polynomials and derivatives in this 37-minute mathematics lecture. Delve into the composition and translation of polynumbers, leading to the introduction of Taylor polynomials and sub-derivatives. Challenge traditional calculus concepts by examining the subject without relying on limits or real numbers. Engage with exercises to calculate Taylor polynumbers and gain a fresh perspective on mathematical foundations. This lecture, part of the MathFoundations series, aims to reshape understanding of calculus, drawing inspiration from the works of Euler and Lagrange.