Regarding Peano's Axioms According to the Wikipedia entry on the Peano axioms: "the number 1 can be defined as $S(0)$, 2 as $S(S(0))$ (which is also $S(1)$), and, in general, any natural number n as the result of n-fold application of $S$ to $0$, denoted as $S^n(0)$." (where $S(n)$ is the successor function). The issue I have is with the statement "n-fold application". With these axioms, we are trying to define what the natural numbers are axiomatically. Therefore, we cannot use the natural numbers to define themselves (or so I think). However, using something like "n-fold application" within the axioms--where n is a natural number--is doing precisely that, is it not (using numbers to define what numbers are)?

You have identified the circular nature of definitions in mathematics. When you are building up the fundamental concepts in mathematics, there is no lower level of foundation on which to build. So you have to use "naive mathematics" to boot-strap the initial structures. (When I say "naive", I mean innate or in-born, or whatever you learned before formal mathematics.)

In this case, the number $n$ is used in two senses: the naive sense and the axiomatic sense. The wikipedia explanation is using naive language to explain formal definitions. As André Nicolas has written above, it is an "informal comment", which is sometimes viewed as metamathematical. Thus the formal $n$ may be explained as $S^n(0)$, where the superscript $n$ is an informal $n$. It is a constant struggle in the foundations of mathematics to try to keep the formal and the informal separate.