This is a perennial and extremely vexing question about which there continues to be great debate. You might find this essay in the Stanford Encyclopedia of Philosophy to be of value.
This is a follow up on http://www.amherst.edu/askphilosophers/question/51, whether the mind can understand how the mind works. In Alexander George's response, he said, "it 'follows' from Gödel's result that there is some basic fact about our minds that we cannot ever know, that we could not in principle access." But is that fact necessarily about the how the mind works, or could it be about some other aspect of mind? As a second question, if we were told what it was, we might not be able to prove it for ourselves, but what would keep us from understanding it in its stated form?
Very loosely and given all the assumptions of my original response, the "basic fact about our minds" in question is the fact that the rules that constitute our minds do not produce conflicting results. Is that a fact "about how the mind works" or "about some other aspect of mind"? That's too vague a question to answer, I think. The fact in question is (given all the assumptions, etc.) a basic property that our minds possess but one that we could not know that they possess. We could "understand" this property, in the sense that we could formulate the claim that says that our minds possess that property. But our minds would not have the means to establish that the claim is true.
Given that 'mental distress' will afflict at least one in seven of us, and as many as one in four (all according to contemporary extrapolations of evidence), and that the spectrum of analyses pertaining to 'mental health' is riven with contradictory perspectives, conceptual muddles, and what even a cursory examination would reveal as potentially harmful (to an individual) prescriptions, and the crossed borders of morality, scientific objectivity and 'spiritual' domains, why is there so little on a 'philosophy of mental health', and where should such a philosophy begin?
There's a link to the Stanford Encyclopedia of Philosophy toward the bottom of the "Related Sites" list on the right.
If we built a computer that could analyse our minds, and it figured out how they work and explained it to us, would we be able to understand?
The great Austrian logician, Kurt Gödel, proved a remarkable theorem in 1931 that he thought was relevant to this question. His theorem wasn't about minds, but with a bit of license, it could be taken to have some implications about them. For instance, this one: Assume our minds are like powerful computers, devices that manipulate symbols according to well-defined rules. Assume , moreover, that these rules are consistent with one another, that is, that they do not yield conflicting results. Then it "follows" from Gödel's result that there is some basic fact about our minds that we cannot ever know, that we could not in principle access. I suppose you might put it, as Peter Lipton did, by saying that that basic fact is "too difficult" for us to understand. But just a slightly more powerful mind would be able to grasp the fact in question about our minds! And that slightly more powerful mind would, in turn, fail to be able to grasp the same basic fact about its own functioning! So, if ...