Comment by kcexn

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.)