I'm of the opinion that math is something between the two, really. At the end of the day, understanding math lies both in being able to construct and follow rig

I'm of the opinion that math is something between the two, really. At the end of the day, understanding math lies both in being able to construct and follow rigorous formal declarative proofs and being able to visualize and abstract what those rules have unearthed. Generally conversation switches rapidly between those two modes using each one in its forte to move the discourse along. I won't argue that math is often taught very poorly, but I also want to bring up that most people only want unconnected heuristics. Number theory and algebra are far simpler and more beautiful than add/subtract/multiply/divide, but they're also not very useful in a grocery store. Most people are script kiddie mathematicians at best. The final thing I want to mention is that much of math is discussed at a level that is very far from "formal" but still rigorous and constructive. At this level, there is exploration whereas the formalities tend to exist to fill in the gaps and understand all edge cases. Rigor seems to me to have less value in formal language manipulation and more in complete understanding of the assumptions and conditions required to uphold your theories. So yeah, math can be formalized, but a lot of it exists as a notational convenience sitting upon oceans of convention that let us rapidly invoke and discuss abstract notions. This is why nigh every paragraph in math begins with affixing some temporary, useful notation to the abstract concepts you've introduced.