This question is slightly subjective, so I'm not sure this is the right place to ask it.
Algebra and analysis are the meat and potatoes of modern mathematics. I have good intuition for discrete math, so I tend to lean on shaky metaphors to think about what other mathematicians would regard an mundane things. When I did undergrad real analysis, I thought of some definitions in an algorithmic challenge-and-response type of manner. E.g. the meaning of $\lim_{n \to \infty} s(n) = L$ was if you want to calculate $L$ with $d$ digits of precision there is some function $n(d)$ such that $s(n(d))$ will give you $d$ digits of precision. It's almost an engineering at this point---figure out how $n(d)$ works and prove this property.
I'm reviewing measure theory and I'm not building intuitions. For stuff like $L^p$ spaces I keep falling back to finite dimensional vector spaces. E.g. for Rietz representation theorem I'm thinking of a bounded linear functional $F:(X \to \mathbb{R}) \to \mathbb{R}$ being expressed as $F(f) = \int f g$, I'm actually playing with functional $F:\mathbb{R}^n \to \mathbb{R}$ begin written as $F(f) = (f, g)$ (where the parenthesis are the dot product and $f$ and $g$ are vectors). For other stuff, when I'm working through exercises it's a slog and I'm just trying modify proofs in the section for the exercises.
Has anyone written an article or book on trying to see analysis through a finite lens. Not in a foundational way (like pushing a Finitist agenda) but for the purpose of explaining the field?
