In this video we venture into a range of topics, from the nature of the continuum, to the paradoxes of Zeno, to an understanding of some of the consequences for mathematics in the shift from geometry to arithmetic that flowed from the Cartesian revolution. As we let go of the real numbers, we must prepare ourselves to appreciate that the arithmetical view of geometry does not always exactly fit with our physical intuition. Our biology constrains us in important ways, and so guides our thinking down certain paths, for better or worse.If we demand that mathematics adheres to our biological orientation and physical intuitions, then we can be led astray. Such is the reason why modern pure mathematics has lost its way logically, with its unreasonable insistence that there is a theory of real numbers.
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.