[LolWolf]

Category theory is fine, I guess, but there's no need to invoke a sledge-hammer when talking about 99% of most work here.

It's a little like saying, yes, fine we can reference the hyperreals to derive infinitesimal calculus in a beautiful way, but it's mostly abstract and highly involved, even to just understand the foundational aspects (which one may argue is the case for normal calculus, but the notion of local linearity I believe is much more obvious than an extension to the reals). Though it's a nice academic exercise, which may even yield some insights, it's rather obscure and unintuitive for almost all purposes.

This is the way I view category theory and functional programming; though I'd love to be corrected if I'm wrong. There's no need to invoke almost all of it, since most things can be expressed quite clearly in the language of set theory need there be rigour.

[eastWestMath]

But infinitesimal calculus is all about local linearity, in the analytic formulation you're taking the closest linear approximation to the function at that point. Using the dual numbers approach, there is an infinitesimal (nilpotent) region around each point which is linear.