To get started, I wanted to give a solid example of the type of ideas I'm talking about. One area that I find particuarly interesting (because it is so relevant to us, and yet so mysterious) is neurology. In an ironic twist of fate (or perhaps "fate" is not the right word), we do not understand how our own minds work. I have a lot to say about neurology and the kinds of questions we can ask therein, but I wanted to start with just a simple question:
Let us define a function f:(B,S)->B that takes the current state of the brain and any stimuli (eg, sight or sound) to a new state of the brain. Given any state of the brain, is there a set of stimuli that could lead to any other desired brain state? Put more mathematically, for all b and b' in B, does there exist an s in S such that f(b,s) = b'? It is hard to wrap your head around what this means, exactly, but you can see the weirdness of what I'm talking about.
For example, is there always some set of stimuli that will make me think of chocolate chips and hamburgers, no matter what mood or mindstate I'm in? You might say, yes of course, if you see chocolate chips and hamburgers, or taste them, you will probably be thinking about them. But what if we ask a trickier question (that is still a specific case of the above generalization)? One state of the brain could include "understanding the Hodge conjecture". I will be the first to admit that my brain has never entered this state. But given the right stimuli (no matter how strange, or alien to our human ideas or perceptions), could I suddenly understand it?
You will notice that this post was not very rigorous, and while I could have been much more specific about a lot of these things, I just wanted to give any readers a flavor of the kind of things I think about.
Looking forward to hearing your ideas!