The charitable version of this is that to reconcile all the holes, we in fact need radically new and different mathematical underpinnings that aren't currently on the horizon. I don't know how that could be true; certainly any new foundation would have to reduce to something very like the current theories under already-studied conditions. If it is, though, we might be on a really big local maximum, and the path off of it might look really weird and nonsensical for a long time (which is why I can't quite bring myself to fully dismiss Stephen Wolfram, for instance :D).