The role of axiomatics in mathematics is a highly contentious one. Originally the term always referred to Euclid, and his use of the term to mean `a self-evident truth that requires no proof '. However in modern times the meaning of the term has shifted dramatically, to the idea that an Axiom is `a convenient fact that we assume'. This casts considerable doubt on the validity of the usual claim that `Mathematics is built on Axioms", which these days appears more and more as a religious position rather than a scientific one. Is that what we want our subject to be?? Your belief system as opposed to my belief system??In this video we discuss this shift in meaning and its consequences when trying to set up a theory of real numbers. We will be discussing this important issue further when we get around to critiquing `modern set theory'.
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.