Full Idea
The Peano Axioms are categorical, meaning that they describe a unique structure.
Gist of Idea
The Peano Axioms describe a unique structure
Source
David Bostock (Philosophy of Mathematics [2009], 4.4 n20)
Book Reference
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.103
A Reaction
So if you think there is nothing more to the natural numbers than their structure, then the Peano Axioms give the essence of arithmetic. If you think that 'objects' must exist to generate a structure, there must be more to the numbers.