Here is a paradox for cognitive neuroscientists: We're trying to understand the brain with the very mental resources that are afforded by our brains. We hope that the brain is simple enough that we can understand it; but it needs to be complex enough for us to be able to understand it.

This is not completely unrelated to Gödel's theorem, which states -roughly- that in any sufficient complex formal system, there exists truths that are inaccessible to formal demonstration. Strictly speaking, Gödel's theorem does not apply to the brain because the brain is not a formal system of rules and symbols. Still, however, it is a fact that the tightly constrained structure of our nervous system constrains the thoughts that we are able to conceive. Our mathematics, for instance, is founded on a small set of basic objects: a number sense, an intuition of space, a simple symbol-manipulation system... Will this small set of representations, crafted by evolution for a very different purpose, suffice to understand ourselves?

I see at least two reasons for hope. First, we seem to have a remarkable capacity for constructing new mental representations through culture. Through metaphor, we are able to connect old representations together in new ways, thus building new mathematical objects that extend our brain's representational power (e.g. Cartesian coordinates, a blend between number and space concepts). Second, and conversely, Nature's bag of tricks doesn't seem so huge. Indeed, this is perhaps the biggest unanswered question: how is it that with a few simple mathematical objects, we are able to understand the outside physical world in such detail? The mystery of this "unreasonable efficacy of mathematics", as Wigner put it, suggests a remarkable adaptation of our brain to the structure of the physical world. Will this adaptation suffice for the brain to understand itself?