I find that most arguments that I have had with people usually end with them implying that their epistemology is rooted in something that is "good" or "reasonable" or "close enough" without any explanation of why it is that their chosen way of looking at the world must necessarily be beneficial. Indeed, you have pointed out that maths is a source of "important and valuable things", which I could agree with, but only with the understanding that our agreement is bound to the reality defined by our joint observations, and not a universal truth.
My point was not about choosing axioms, but about incompleteness, undefinability, uncomputability, etc., which cut short any attempts to bless a particular set of axioms as uniquely correct. You are only seeing the surface level so far; I assure you that it all deconstructs. (Be careful when looking directly at The Void.)
Finally: Where does meaning come from when having dialogues? Hofstadter explains it well: We constantly exchange deeply-coded messages with recursive layers of meaning, trying to reflect the symbols in our mind into words which we think will reconstruct those symbols in the minds we talk to. The "mooring of their meaning" is a fascinating illusion based on the inherent difficulty of stepping outside ourselves to examine where our own sense of meaning comes from.