Why do we need arithmetical operations instead of teaching arithmetic based on the successor function?

Question

Since we can construct the set of natural numbers off of the peano axioms with the operations of addition, subtraction, multiplication and division following, why do we even need 4 arithmetical operations? Since the successor function is "modelled" off of counting as we know it as babies, it is quite intuitive for many people. Why then don't we just teach all of arithmetic based off a axiomatic, intuitive, and quite "primitive" notion of the successor function, and why do we specifically need 4 arithmetic operators? (Specifically where even the basic notion of addition of natural numbers is hard to put into words, without a lengthy exposition which a young person would have a hard time understanding).

Answer 1

That is, in effect, what the Peano definitions of the arithmetic operations do: they build them from the successor function - e.g. the recursive definition for addition is

$$a + 0 := a$$ (base case)

$$a + S(b) := S(a + b)$$ (recursion case)

It becomes clearer if one notes that effectively almost by definition,

$$b = (\underbrace{S \circ S \circ S \cdots S}_{b})(0)$$

so that

$$a + b = a + (\underbrace{S \circ S \circ S \cdots S}_{b})(0) = (\underbrace{S \circ S \circ S \cdots S}_{b})(a)$$

which says that "$a$ plus $b$ works by adding $b$ single things one at a time to $a$", and if you want to show a child, show hir stacking little blocks of hir favorite type together on top of a given tower of blocks, one block at a time, and then the remaining tower has a number of blocks that equals the sum of the two amounts in the preceding towers.

The reason we have the operation of addition is because it very quickly gets awful to say "two, then one more, then one more, then one more, then one more", and much better to say "two plus four". And better, because it generalizes beyond the simple case of natural numbers.