In this video we ask the question: how do standard treatments of calculus and analysis deal with the vexatious issue of defining real numbers and their supposed arithmetic??We pull out a selection of popular Calculus and Analysis texts, and go through them with a view of finding out: what exactly is a real number? All the books I examine are excellent books---aside from their treatment of foundational issues, where we see that they mostly fall clearly short.We look at Calculus texts by Steward, Sallas Hille and Etgen, Rogawski, Courant, Spivak, Caunt, Apostol, Keisler and Adams, and Analysis texts by Spiegel, Apostol, Royden, Kolmogorov and Fomin, and Rudin.This video really should be an eye opener to students of mathematics. Yes, it is possible to challenge the standard thinking, and the mathematical world need not collapse. Admitting current weaknesses, and the lack of acknowledgement of them by the Academy, is an important step in moving forward to a newer, better 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.