I have always wondered if we would be capable of writing down the mechanisms that power our thoughts. I think that this was one of the ideas that bubbled up from reading Godel Escher Bach many years ago. Is it possible for us to express the machine that makes us using the outputs of that machine in the way that it's not possible to write second order logic using first order logic.

Of course, also there are processes that are not expressible as computations, but those of these that I know about seem very very distant from human thought, and it seems very very improbable that they could be implemented with a brain. I also think that these are not observed in our universe so far.