There is a good reason why pure mathematicians cling so tenaciously to the idea of real numbers. They provide us with the ostensible `values' that lengths, areas, values of functions and solutions to equations seem to require. But is this all really just a dream??In this video we have a new look at these notions, with a view of examining whether they really do support `real values', or whether perhaps they are intrinsically approximate notions. To motivate the discussion, we go back to a crucial calculation of Archimedes.
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.