The natural numbers are 'natural' because they are definite quantities that can be used for counting.

Taylor expansions about a point of a function requires that the function has a derivative defined at that point.

The derivative itself is the point at which an infinite sequence (say, of incrementally closer approximations) converges.

So derivatives and Taylor series are really more of an arbitrary precision approximation of a value rather than a concrete exact quantity.

Arbitrary precision approximation just happens to be a very elegant way to model the physical world around us.

For truly exact solutions, you still have to work with the naturals (and rationals, etc.)

But you can't really have chemistry without working with natural numbers of atoms, measured in moles. Recently they decided to explicitly fix a mole (Avogadro's constant) to be exactly 6.02214076×10^23 which is a natural number.

Semiconductor manufacturing on nanometer scales deals with individual atoms and electrons too. Yes, modeling their behavior needs complex numbers, but their amounts are natural numbers.

I agree. Had humanity made turning the more fundamental operation than counting that would have sped up our mathematical journey. The Naturals would have fallen off from it as an exercise of counting turns.

The calculus of scaled rotation is so beautiful. The sacrificial lamb is the unique ordering relation.