Non-computability.

According to Roger Penrose, humans can perform non-computable feats, such as dealing with Gödel questions. He uses this as a foundation to claim that the human mind cannot be expressed in terms of classical processes, and as such must be party to the only other (known) game in town: Quantum Mechanics.

Now, I haven't had the patience to sit through all of his arguments yet, though I slowly make progress. My understanding is that a large part of his stance is that an algorithm cannot usefully deal with a Gödel question, or equivalently, with the halting problem, while a human can.

My objection to this is that such problems always demand a certain quality of response when asked of UTMs: failing to respond forever is not acceptable as correct, nor is providing any response other than one that yields a truth when taken in combination with the question. This much is fine, however, when it is time for the human to answer, he is permitted the liberty of rejecting the question on the grounds that it is inherently unanswerable.

Obviously I am interested in artificial intelligence, and also find his assertion to be simply a self-serving one with a contrived philosophical backdrop for foundation. If anyone knows of, or can think of, a more sophisticated argument than the one above (or expose my flaws in my assessment of it) I would like to hear it.

Apologies for bringing up a largely philosophical question, my only excuse is that I cannot trust any other board with the question.

>>4
No, look, the onus is probably on you here, for positing the solution, but I'll go ahead anyway.

A common demonstration of the halting problem is to suppose that a candidate Turing Machine (TM) for solving the halting problem is modified to exit if it determines that the TM it is analyzing will not halt. If it determines that the analyzed TM will halt, it goes into an infinite loop. This is a simple enough modification that any /prog/lodyte could muster.

Obviously, when applied to itself, the candidate fails and must be rejected. This is true of any candidate, and moreover, any candidate without modification must also be able to accurately evaluate the modified candidate. So all unmodified candidates fail for this case.

To show this applies to all finitely expressible TMs, presume the existence of a finitely-expressible candidate TM, and make the modification, which shall not introduce an infinite requirement. In this case, it is as simple as the case above: all candidates are unsuitable.

If you chose to reject this on the grounds that no finitely expressible TM can suitable to serve as a candidate, then it is as simple as that: there is no finite-resource scenario for solving the halting problem.

Q.E. friggin' D.