Although it is primarily a matter of taste, recursive definitions can also be recast in an inductive setting. As a first example, set $1!=1$ and whenever $k!$ has been defined, set $(k+1)!=(k+1)k!$.
Let $m$ be a positive integer. Then set \begin{equation*} m\cdot1 = m\quad\text{and} \quad m\cdot(k+1)=m\cdot k+ m \end{equation*} You should see that this defines multiplication but doesn't do anything in terms of establishing such familiar properties as the commutative and associative properties. Check out some of the details in Appendix B.