I've had discussions with people before who make the claim that consciousness is somehow going to be forever beyond the realm of science. Here's a Bloggingheads.tv discussion between Eliezer Yudkowsky and Jaron Lanier. About 25 minutes in, this is exactly the kind of point that Lanier tries to make:
If I'm understanding his point, he's saying that he doesn't understand consciousness, but that an understanding of consciousness will not be possible under the assumption that it is a product of physical things (like neurons) carrying out their function in the context of a physical system (like the human brain).
Yudkowsky calls him on this, by basically asking how you can make claims about how something is or isn't going to be understood if you don't understand it. It's a great question, but Lanier does what he does throughout the discussion, which is either laugh and move on to another point or make some kind of ad hominem slur against Yudkowsky by calling his adherence to science a kind of religion.
Lanier ironically states that making such groundless, pessimistic claims somehow makes him a better scientist. Huh?
Look, everything in the world was mysterious before it was explained. Lightning and thunder, for example, were very mysterious. Ancient people came up with initial explanations having to do with the actions of supernatural beings. Those explanations turned out not to be very good. We actually started to get somewhere when someone went, "Hey, wait a minute...maybe there's a reasonable explanation for what's going on here that we can understand."
That assumption, and not a hypothesis or theory or experiment, is the beginning of science and the first step on the road to knowledge. If you don't take that step, and automatically assume that either the explanation is supernatural and/or that you will never be able to understand it, then what you have done is guarantee that you will never understand it because you've given up before you even tried.
The history of science is one in which we have made progress in our understanding of the world by making the default assumption that things are the result of natural processes that work in orderly ways according to principles that we can figure out if we work hard enough.
It may very well be the case that there are hard limits to human understanding, and that there are things we are not going to be able to figure out. The nature of the universe may be one such question, whether or not there is a single frame of reference or some kind of multiverse, why the universe as we know it is expanding and whether or not it is a single case of such an expansion or one of many cycles of expansion and collapse. A grand unified theory of physics that reconciles quantum mechanics and relativity may be beyond our understanding. And so might consciousness. But the fact is, we have barely even begun to try to systematically understand these things. A few rare people have pondered such questions for millennia, but organized science as an institution has only really picked up speed in the last 150 years or so, which is really a very small amount of time.
So isn't it just a tad early to throw in the towel?