I was actually intent on asking this question in a philosophy forum because it relates to methodology of mathematics itself more than the technical operations of doing math, but it doesn't hurt to leave no stone unturned, i guess.
To begin, i know that a mathematical proof is an inferential argument for a mathematical statement, and i know that the argument may use other previously established statements, such as theorems; also i know that every proof can, in principle, be constructed using only certain basic or original assumptions (axioms)
but the more you think about it this way, the more complicated it gets. in the case of the question posed, the mathematical axioms that relates to theories with large quantities of numbers in its domain are not always logical axioms, some are non-logical axioms, or "postulates" that are merely substantive assertions about the elements of the domain of a specific mathematical theory.
it feels like hitting a rock to me, at some point you feel like you are dealing with theories with large domain of numbers in a pragmatic way (simply because they work in the smaller domains we need), thus discovering the limitations along the way, rather than actually existing method that has sure certainty in its nature, which i believe is the already accepted norm since Godel's incompleteness theorem was proposed.
But then again, this really makes our ways of calculating confidence in a theory look pretty weak, so to pose it in a less academic way, how are we confident about our confidence?