Rachael pointed out an interesting objection to Searle's Chinese Room argument: the program that the operator in the box is executing can contain no specific information about the operator themselves because the assumption is that the operator is interchangeable. Thus, much like human minds, the box couldn't introspect about its own internal structure.
I've also just stumbled across "process algebra," which this article is claiming has the potential to do calculations that Turing machines are incapable of. I thought the whole idea of a Turing machine was that it could do any calculation whatsoever; but I suspect the trick lies in the temporal and memory constraints imposed by real world calculations. (A universal Turing machine can perform any computation, given infinite time and infinite memory).