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.

Since recursion and looping have been brought up, let us use strict definitions.  A program that will halt on certain input can iterate upon itself as many times as necessary and make however many calculations, including the same calculations, as many times as necessary before finally halting.  Finally halting.  A program that does not halt on certain input can iterate upon itself and make however many calculations but will never halt (will never return).

Any heuristics that impose a time limit or a loop limit on this problem defeats the purpose of a Universal Turing Machine that definitely proves whether any valid input on any computable program lets that program halt.  Deterministic finite-state automata don't work that way and such assumptions escape the thought experiment entirely by allowing a state which we don't consider a part of the permit solutions - "give up."

If I may conjecture, saying that we allow "give up" seems to imply our intended result anyway.  We can not build a Turing Machine that can satisfy the halting problem.