Perhaps "math" is overloaded to sometimes refer to the actual doing of mathematics and at other times to the act of communicating mathematical ideas - which is

Perhaps "math" is overloaded to sometimes refer to the actual doing of mathematics and at other times to the act of communicating mathematical ideas - which is where symbols/logic/rigour begin to feature. As for "intuition", I usually seek refuge in Jeff Raskin's theorem "intuitive = familiar" - i.e. you don't lose the content of a statement made using the word intuition if you replaced it with the word familiar .. but now suddenly it is less "mystical" and clearer. Plus it points. To the subjectivity of intuition. - that what is familiar to one may not be familiar to another and so there are different "intuitive" explanations. So "intuitive understanding of topic" = "understanding a topic based on what you're already familiar with". Explains why quantum mechanics doesn't feel "intuitive". There is nothing in our experience that makes the phenomena "familiar" to us.