Q:

Formal systems and interpretations for numbers

In my book (Hodel's Intro To Mathematical Logic), we are given several examples of formalized mathematical theories such as group theory, Peano arithmetic, etc. But I've had this ongoing confusion: the formal system can generate theorems, but then the book goes on to talk about particular interpretations that we can provide for the symbols of the language. For example, we are given the standard interpretation N of PA for Peano Arithmetic where we're given a domain of natural numbers, a 0 interpreted as the number zero of the natural numbers, etc.
My confusion is that numbers don't exist 'out there' like tangible things do. It's easy for me to see how a sentence Pa is true in an interpretation where a=John and P='is a person', but not so much for numbers.
So: What does the book mean when we provide the standard interpretation (i.e. what is it that we are interpreting)?  What do we mean when we speak of "truths of arithmetic" that lies outside of the formal theory of Peano Arithmetic? Are the numbers (and their operations) defined outside of the system or something?
p.s. this is especially confusing considering I would call myself something like a fictionalist, or formalist, or something of that nature.

A:

You are correct that until we know there are numbers, we have a formal theory but not yet any domain where the theory can be interpreted.
There are two ways to go with this. 
1) It's clear that we all know what the natural numbers are. Even if we don't have a formal theory about them, it's clear that they satisfy PA and are therefore a domain in which we can interpret the axioms and theorems of PA. 
2) We can formalize the axioms of set theory, say Zermelo-Fraenkel (ZF). Then the concept "exists" means "can be shown to exist from the axioms of ZF." Then we can say that the empty set stands for zero, and the set containing the empty set is 1, and so forth. The objects of this construction are known as the Von Neumann ordinals. 
http://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals
Now we can verify that the numbers 0, 1, 2, ... as defined by Von Neumann, satisfy PA; and therefore we have a set in ZF that can serve as a domain of interpretation for PA. 
The same thing is often done pedagogically with the real numbers. The usual procedure is to list the properties of the real numbers such as the commutativity of addition, the distributivity of multiplication over addition, and so forth. That's good enough for calculus class and lets you do everything you need to.
In a math major class called Real Analysis, they'll show that you can start with the rational numbers, and construct a model of the real numbers using Dedekind cuts or Cauchy sequences. They show that to the students once, and from then on everyone forgets about it and just uses the properties of real numbers. 
In practice it turns out that most of the time nobody cares about the ontology of mathematical objects. All you need is their properties. 
But ultimately you're right, if we're going to be logically comprehensive it's not enough to write down the axioms of PA; we have to show that we can formalize something that satisfies them; and that's where the Von Neumann construction comes in. 

