Working mathematician does derive his ideas from axioms, it is ridiculous perception of mathematical process. And axioms are not logical consequences of sensible assumptions, it is try and fail process, long historical process. Intuition in mathematics has nothing to do with common sense intuition. Respectfully.

In abstract mathematics, one can choose any axioms one wants. To derive all kinds of consequences/theorems.

But respectfully, ZFC came about because of logical paradoxes that couldn't be accepted as consistent.

You can create a new number system and derive all kinds of consequences but the truth is, most mathematicians care more entirely about prime number theory on the naturals that are entirely based on counting.

Most of modern math is based on the natural numbers. You can't remove all intuition. The thread that holds our love for math is also the same one that tells us we are exploring consequences that tell us a dearth about our universe.