There are axioms, and then there are well established methods of working with them. It is clear that these methods are nothing but logical operations (rules of manipulation of symbols) on previous statements.
Let us say we are required to prove a particular result. All we have at our disposal are axioms, and rules of manipulation. What I wish to ask is the following:
Since all we will be doing is just logical operations on axioms, it means that the result resides in the axiom itself. It therefore seems that: Axiom dictates what is the case, and also what is not the case.
What value does the logical operation adds to the whole process? Why axiom itself does not reveal the status of conjecture (true or false)?
In general, what happens when a mathematical proposition transitions from one expression to another?