| > Probability has a connection to Lie groups because it's fundamentally analytic ("continuous"). But you haven't explained how you make the connection to the dual numbers. Are you familiar with Chu spaces? > But just because you're working with continuous spaces doesn't mean you've engaged the duals. It generally means you're using the reals. They are not just continous spaces, it's a pair of a discrete space and a continous space that are directly connected. You never work only with one of them at once. You manipulate things in smooth space to solve things in the discrete space and vice versa. > This gets to the heart of what I'm saying - if I wanted to be flippant I could have said the real numbers, or continuity, or analysis, etc are at the heart of so many distinct subfields of mathematics. It doesn't mean quite a lot. Analysis is too general and also much higher conceptually. Also, you need to be looking at constructive mathematics to really capture duality. Also analysis is unusuable for a lot of problems that duality is useful for. For example the Rust borrow checker is based on linear logic, a logic that reifies the concept of duality. No one has ever used analysis to build a compiler. |