To give an example - here's a problem a Turing machine (e.g. a computer) can't solve:
A) Suppose you write a program whose purpose is to examine another program and decide whether the program halts. Let's call this program H. If a program halts, H does not halt. If the program does not halt, H halts.
B) Feed H to itself. Now there are two possibilities:
B.1) H decides that H halts, in which case H does not halt. Oops!
B.2) H decides that H does not halt, in which case H halts. Oops again!
Now, a computer can't solve this problem. My question is, can anyone?
While I personally am quite anti-religious, some religious people use intractable problems of this nature to suggest that the 'unknowable in principle' exists...
Peter --
In some later post, there is mention of Hegel. Going back to the Turing machine problem, I think this is re-formulation of the set theory paradoxes. Whether it maps to the set of all sets, or the set that contains itself as a proper subset, I don't know. But this Turning machine problem has to be a variation on one of those.
This is not a proof of the unknowable in principle. Here are several reasons why not.
The base line assumption is that knowledge can be subsumed in a logical system. But Kant somewhere says, that while all knowledge arises from experience, all knowledge is not necessarily contained with in experience. You can turn this around and say that while the activities of the mind (logic) can account for some of our knowledge, they do not necessarily account for all of it, and certainly not all of which arises from experience.
You can keep going along this line. We more or less assume that the knowledge that arises from experience is expressible as some form of empiricism. But then there is a further assmption that takes empirical knowledge to be synonymous with scientific knowledge. This assumption is that these are completely synonymous with one another. There are plenty of things I know, that I have discovered from experience that are not expressible as scientific knowledge and yet are empirical in nature. And, there are plenty of other things I've discovered from experience that are not entirely empirical. In a general domain of knowledge, then empirical knowledge contains some scientific knowledge, but not all, since at least some of what is called scientific knowledge is derived from activities of the mind independent of experience.
But to return to the main example here of the Turning machine. There are a whole collection of these paradoxes and I think the more interesting of them leads to cosmological questions that have shaped the way we think about the origin of universe, the origin of conscieousness and most of the other big questions. There is something profound wrong headed about all these. Of course I don't know what is wrong, but something is, because so far the conclusions and answers that scientific work has come up with are ludiciously absurd, to my sensibility.
This is not to say there is a religious or mystical answer either, since these forms of thought and belief are still further down the list of ridiculous ideas.
gotta go to work
Chuck Grimes
ps. sorry about that last empty post. cli is a quick way to make big mistakes.