Here's something I'd like to figure out. When philosophers ask "Does mathematics need new axioms?", what is the is the intended task, such that they are asking whether we would need new axioms to accomplish it?
Here are some possibilities:
-to know all mathematical truths (well, we can't do that, with or without new axioms)
-to formally capture all our intuitive judgements about mathematics (there are familiar putnam vs. penrose reasons for thinking we can't do that either)
-to formalize some particular body of generally accepted mathematical reasoning, where everyone agrees on what's a good argument, but this can't be captured by logic plus the axioms we currently accept, and having a formalization would be practically helpful.
-to be in a state of believing all propositions which we are justified in believing.
It seems to me that, there's a great danger of launching into the debate about whether "math needs new axioms", and taking a position based on whether e.g. you like or dislike set theory, without having any clear sense of what you are claiming that we do/don't need new axioms for. Hence, I'd like to get clearer on different senses the question can have, and which one(s) are at stake in typical philosophical discussion.