I'm a programmer by trade, so my knowledge of mathematics as a whole is limited by what I happened to learn doing my undergrad in CS, so a lot of the mathematic-specific examples in this post were lost on me.

What I found particularly insightful here is applying this mathematical notation to programming languages and their syntaxes. His notation described as ``` Notation:{well-formed expressions}→{abstract objects in 𝑋} ``` isn't too far off from what most programming languages implement at some level.

As a result of this, what properties do programming languages share with mathematical notations, and why are some languages deemed more "expressive" than others? How much does the "expressiveness" of a certain language in a domain lead to better understanding of the abstractions beneath them?

To answer your question; I don't think this post explicitly increased my understanding of notations (especially not in the context of mathematics), but rather led me to ponder the importance of them in communicating & extending abstract concepts effectively across domains.