Comment by JdeBP
6 days ago
The entirely opposite perspective is quite interesting:
The "natural numbers" are the biggest mis-nomer in mathematics. They are the most un-Natural ones. The numbers that occur in Nature are almost always complex, and are neither integers nor rationals (nor even algebraics).
When you approach reality through the lens of mathematics that concentrates the most upon these countable sets, you very often end up with infinite series in order to express physical reality, from Feynman sums to Taylor expansions.
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.