Towards a New Mathematics and "Types versus Sets" | Sociology and Pure Maths | N J Wildberger

Video Description

What would it be like to have a Pure Mathematics that was completely logically airtight --- by having every single notion clearly encodable on a computer? This would be a huge departure from the current state of affairs, which is anchored in modern set theory with ever present claims about infinite objects and infinite processes which cannot be encoded on any computer, even in principle. Box Arithmetic is a new approach to the Foundations of Mathematics. It proposes to build up the entire structure of Pure Mathematics using only repeated constructions from nesting copies of the simplest possible entity: an empty box or multiset. In my book Algebraic Calculus, being written with Anna Tomskova, we use this as a framework for setting up arithmetic, algebra, linear algebra, and eventually analysis. No "infinite sets" or "completed infinite processes" are required. In this video we explore the crucial distinction between talking about mathematical "types" and talking about mathematical "sets". These are actually separate notions that we need to keep apart. How far can we go with Box Arithmetic? To get a sense of it, head over to Wild Egg Maths YT channel, where the playlist Intro to Algebraic Calculus goes into more detail.