Re intuition, yes, it's being said by notables (Sir Martin Rees included, if I'm not mistaken) that the human mind may be quite incapable of grasping concepts beyond those which served our evolution. It follows that the process of discovery should gradually unveil realities that are alien to our comprehension, and further, that it may unearth signposts that are completely invisible to us.

Re mathematical reality, you are surely right; and I think the mathematicians are fully aware of the fact that maths can describe not only what exists, but also what doesn't exist. Observation and experiment are the verifiers.

"Time is what prevents everything from happening at once" - John Wheeler