How do you understand $a\times1=a$ in Peano axioms?

Question

How do you understand $a\times1=a$ in Peano axioms?

This should be understood as replacing $1$ with $a$? For example, when we multiply $3$ by $2$, that means: replace each $1$ in the number three with $2$?

Is it just a replacement or is it something else?

Answer 1

The replacement simply seems to be distributivity in disguise. Consider $$a\times b=\underbrace{(1+1+...+1)}_{\text{a times}}\times b=\underbrace{(b+b+...+b)}_{\text{a times}}$$Thus, each of the $1$ was 'replaced' by $b$ by simply distributing over $a$. The distributivity of multiplication over addition on $\Bbb N$ can be established using Peano Axioms.

Answer 2

If I remember well, multiplication is defined by induction $0x = 0$ and $(n+1)x = nx + x$. The first step of the recurrence gives $$ 1\times x = 0x + x = 0 + x = x.$$ Clearly you can either define the multiplication on the other side, if you wish to prove it directly in the form $x \times 1 = x$ or by proving (induction) that the multiplication is commutative.