Before the invention of the calculus, there existed no mathematical tools to deal with phenomena in a constant state of change. The calculus made it possible for the very first time to build non-empirical mathematical models of non-trivial physical phenomena. Calculus was "new math."
Before the invention of Fourier analysis, the question of heat transfer was analytically intractable. Fourier opened up frequency space and paved the way for most of modern engineering, electronics, and communications theory. Fourier analysis was "new math."
A physical singularity like a black hole poses an analytical problem because the analysis "blows up" as you approach the singularity, just like in the f(z) = 1/z example above. At a mathematical singularity, we say the solution "exhibits a discontinuity" and "approaches infinity," and they DO.
At a physical singularity, our mathematical solutions do the same. We have an innate sense that there must be SOMETHING going on in there... but it is the nature of a gravitational singularity that it is cut off from observation by the rest of the universe at its event horizon, and it is the nature of our "innate sense" of things that it is often flat-out wrong.
Since there is no test we can perform to test any conjecture we have regarding the interior of a black hole's event horizon, and since the limitations of our mathematical tools prevent us from even MAKING a conjecture regarding conditions at the singularity, the most intellectually honest statement we can make regarding those conditions is that we have no idea.