I'm not sure that's the case. He's not saying that the mathematics doesn't represent the physics. He's saying that in order to apply the mathematics we have to

I'm not sure that's the case. He's not saying that the mathematics doesn't represent the physics. He's saying that in order to apply the mathematics we have to have some empirical model to which the axioms of the mathematical system applies. As long as that's the case, the mathematics can provide insight into the physical system. But where the mathematical tools break down are in the areas where we don't have any data. From a physical perspective the math, in this case, is moot. We have no data about what happens past the event horizon, so I could just as easily say that it encompasses a space filled with tiny pink unicorns which exhale confetti and it would be just as meaningful. What we know is that something different happens. So there's really no room for either the mathematicians or the physicists to speak with authority on what happens. What we're left with is very well-informed speculation about a really interesting region of space-time.