I would like to write about the problem of "expression", from my own experience. In the 1960s, it seemed to me that, from a commonsensical viewpoint, there should be some way of expressing that
in the above diagram the big square should be the "composition" of all the little squares. Then I found that Charles Ehresmann had defined double categories, which did the job.
The next question was: what is a commutative cube? For a square with sides $a,b,c,d$ the answer might be $ab=cd$. But if we want the "faces" of a cube to commute? For all this to be significant one has to move from sets with a total operation to sets with partial operations!
The point I am trying to make is that one function of mathematics is to develop language for rigorous expression, deduction and calculation, and this may take a while to develop. For example Descartes' notion of a graph of a function is now a commonplace, even common sense; but it may take an intellectual leap to make something into "common sense".
Another example is the introduction of the zero, and Arabic numerals.
Are higher dimensions than $3$ common sense? See the book "Flatland"! (downloadable).
Worth discussing is: have there been revolutions in mathematics? See discussions on the work of Thomas Kuhn on "Revolutions in Science".