Automatic differentiation is something that you get for free if you use dual numbers. Lie theory describes the relationship between the discrete and continuous spaces. Probability has this deep connection to Lie groups.
To give you some intuition (and I'm really rephrasing the stackexchange post above), the only way you can only measure randomness (or generate randomness) is if each draw has a "reference" to some global object that has the global, normalized view of the discrete space.
Are you familiar with Vovk's foundations of probability? That brings you from probability to game semantics. Duality is right next to it.
> Automatic differentiation is something that you get for free if you use dual numbers. Lie theory describes the relationship between the discrete and continuous spaces. Probability has this deep connection to Lie groups.
This...doesn't follow. 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.
What you're showing here is that a lot of things in mathematics can be described analytically (and saying that would be likewise pretty superfluous). 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.
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.
Duality features in a lot of different parts of mathematics, but that doesn't mean you can productively draw connections between dual things in one area and dual things in another. I'm not seeing how you get from dual numbers to Lie groups.
> 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.
