What a can of worms.
It is clear that scientific explanations are formulated using principles that are more powerful (in the sense of information or computational complexity) than the explanations themselves. This, in spite of the evident fact that these principles are informal and often, even usually, make mistakes.
Accordingly, no scientific explanation can give a formally complete account of the causes of scientific explanations.
This is what ultimately is bugging people like Penrose and Lucas. Let me explain or unpack this if I can.
Kurt Godel formulated a version of what I am saying in a dilemma: either there are solvable Diophantine equations that nobody, no matter how smart or infinitely long-lived, could ever solve -- or we are not Turing-limited. People may quibble with Godel's assumptions, but they can't quibble with the actual dilemma.
If there is something to what I am saying (and obviously I think there is), there are alternative situations we might be in. Nature could be super-Turing (i.e., might have computational complexity omega); or, we might not be completely natural.
I see do not know of any empirical or practical way to distinguish between these alternatives. Quantum mechanics is relevant -- not by providing some non-computable functionality to the brain, but rather through non-locality!
Naturally, there is no "formal" theory behind what I am saying -- by the very nature of the problem.
But a number of people have grappled/are grappling with the issue. There are formal negative results, e.g. David A. Wolpert, Physical Limits of Inference [arXiv:0708.1362v1]. Especially the theorems concerning "self-aware devices." As far as I can tell (I am not an expert in this field) Wolpert's results amount to a refined formulation of Godel's dilemma. The Conway-Kochen Free Will Theorem [http://arxiv.org/abs/quant-ph/0610147]
also is relevant, since it depends on the assumption that we are free to choose any experimental setup from a set of possible setups -- but then non-locality infuses all of Nature with the super-Turing complexity (or, 'indeterminacy') of our choice.
Again, if we're actually reasoning when we do science, then our reasoning powers are more powerful than any or all of their formal products. If we're not actually reasoning when we do science, what the hell are we doing?
Finally, last thing I heard, you can't reason if you're not conscious.