We begin PART II of this video course: "Mathematics on trial - why modern pure mathematics doesn't work". This video outlines the case for the prosecution: that modern pure mathematics suffers from: 1. Inconsistent rigour 2. Problematic definitions 3. Reliance on `axioms' 4. Computational weakness 5. Impoverished examples We give some initial orientation to the first two claims, suggesting that contrary to popular opinion, rigour in mathematics has been on a consistent downward trend in the last few centuries, and give an explicit list of problematic definitions, cutting across many areas of modern pure mathematics. In subsequent videos in this series, we will be substantiating the claims made here. We will also be inviting comments and a wide discussion of these highly contentious, but vitally important, issues.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.