I think that I just take issue with the term "hand-waving" as equated to intuition. Yeah it lacked formal rigor, but they had a solid model that applied in detail to the real world. That doesn't come from just saying, "oh well, it'll work itself out". I guess if you want to call that "hand-wavy" we'll just have to disagree.

Euclid tells me otherwise. Rules, no art, no bullshit. Rules. Humanities people somehow never get it. Is not about arithmetics.

Vibe-what? Vibe-bullshit, maybe; cathedrals in Europe and such weren't built by magic. Ditto with sailing and the like. Tons of matematics and geometry there, and tons of damn axioms before even the US existed.

Heck, even the Book of The Games from Alphonse X "The Wise" has both a compendia of game rules and even this https://en.wikipedia.org/wiki/Astronomical_chess where OFC being able on geometry was mandatory at least to design the boards.

On Euclid:

https://en.wikipedia.org/wiki/Euclid%27s_Elements

PD: Geometry has tons of grounds for calculus. Guess why.