Mathematics might be called an art instead of science were it not that the last achievement that it has in view is an achievement of knowing. [Peirce 1976, Vol.III, 527]
What is it that mathematicians do and what is the kind of stuff that they are thinking about? The standard method is the axiomatic one. According to it, mathematical (non-logical) axiom systems capture classes of structures as the models of a mathematical system (a theory). Mathematicians then study these structures by applying logical procedures to derive consequences from these axioms. Importantly, it is not predominantly the deductive kinds of logical consequences but the model-theoretic ones that a mathematician is on the lookout whenever new information about the structures is called for.
The axiomatic method, model-theoretically understood, is about logically establishing which structures are possible and which structures are not according to the axioms. The axiomatic method is not about, or at lea...