Math ed runs to applications. They are hard wired now to say applications. It is their Pavlovian response.
What is math good for? Applications.
Why am I learning this lesson? Applications.
Never the structure of math is itself the reason to learn the lesson. The vision of math is the structure of math not the applications. Cathedrals rise up to the sky not to symbolize applications but aspirations.
The Peano Axioms are the real structure and thus the real meaning of addition and multiplication of whole numbers. That translates to the meaning of place value and its operations.
Arithmetic is not mere grunt work. It is creation.
The chain of whole numbers has meaning and inspiration. It starts from 0 and goes to infinity. Its rules are given by the Peano Axiom. Axiom 5 on induction is what connects 0 to infinity. It should not be hidden but put on a hill top as a shining lamp.
Induction is the axiom of proof. It is the axiom of faith. It is the axiom of connection. It should not be buried as too abstract or hard. It is the very stuff of meaning. It is the very stuff of arithmetic.