When I'm thinking about mathematics, I usually imagine that every sentence in the language of arithmetic is either true or false, in reality. Thus, I imagine that truth comes first. Afterwards come axiom systems that are intended to prove as many true sentences as possible, while not proving any false sentences.
However, in practice we seem to begin with a formal system (that is, we begin with provability), and only afterwards do we define truth. For example, we might start with a formal system, say ZFC, and use it to define N, the set of natural numbers, and then prove that the Peano axioms are satisfied by N; that is, they're true (with respect to N). But notice that we had to begin with a formal system, namely ZFC, before we could talk about truth.
As a result, I am philosophically confused about which comes first, truth or provability. This has practical ramifications. For instance, suppose I wanted to write an introductory book about the foundations of mathematics. Which would I talk about first?
What are the major positions in this debate, and where can I learn more? Note that I am not specifically interested in whether or not mathematical entities have any real and independent "existence" except as a means to working out a "time ordering" or "logical ordering" on truth and provability.
EDIT. I'd like to point something out. In mathematical logic, there's an enormous difference between truth and provability. Truth is a relationship between models and sentences, while provability is a relation between axiom systems (or more accurately, "formal systems") and sentences.
So for example, we can say, "For the standard model of arithmetic, Goodstein's theorem is true." Since this has been proven, it is believed true. We can also say, "From the axiom system known as Peano Arithmetic, Goodstein's theorem is provable." Since this has been disproven, it is believed false.
So when I say, "Which comes first, truth or provability?" what I really mean is, "Which comes first, models or formal systems?"