In a recent post, I commented on Reed Montague's proposal that it may be necessary to account for the limits of psychology when developing theories of physics. I disagreed with his thesis because if we accept that physics is computable and the brain is described by physics then any property of physics should be discernible by the brain. However, I should have been more careful in my statements. Given the theorems of Godel and Turing, we must also accept that there may be certain problems or questions that are not decidable or computable. The most famous example is that there is no algorithm to decide if an arbitrary computation will ever stop. In computer science this is known as the halting problem. (Godel's incompleteness theorems are direct consequences of the halting problem, although historically they came first). The implication is quite broad for it also implies that there is no sure fire way of knowing if a given computation will do a particular thing (i.e. print out a particular symbol). This is also why there is no certain way of ever knowing if a person is insane or if a criminal will commit another crime, as I claimed in my post on crime and neuroscience.
Hence, it may be possible that some theories in physics, like the ultimate super-duper theory of everything, may in fact be undecidable. However, this is not just a problem of psychology but also a problem of physics. Some, like British mathematical physicist Roger Penrose, would argue that the brain and physics are actually not computable. Penrose's arguments are outlined in two books - The Emperor's New Mind and Shadows of the Mind. It could be that the brain is not computable but I (and many others for various reasons) don't buy Penrose's argument for why it is not. I posted on this topic previously although I've refined my ideas considerably since that post.
However, even if the brain and physics were not computable there would still be a problem because we can only record and communicate ideas with a finite number of symbols and this is limited by the theorems of Turing and Godel. It could be possible that a single person could solve a problem or understand something that is undecidable but she would not be able to tell anyone else what it is or write about it. The best she could do is to teach someone else how to get into such a mental state to "see" it for themselves. One could argue that this is what religion and spiritual traditions are for. Buddhism in a crude sense is a recipe for attaining nirvana (although one of the precepts is that trying to attain nirvana is a surefire way of not attaining it!). So, it could be possible that throughout history there have been people that have attained a level of, dare I say, enlightenment but there is no way for them to tell us what that means exactly.