Well, no, human thought isn't so limited, and therefore requires additional physical explanations than have so far been offered. That's Penrose's and Searle's point. The human mind, whatever it is, they say (and I agree) isn't a digital computer. Further, it demonstrably can do things a digital computer can't. Marvin Minsky once gave a 'trivial' problem to a graduate student: to have a robot do some fairly rudimentary takss of pattern recognition and manipulation. No success yet.I'm inclined to say "No, of course human thought isn't so limited," but then, how do we know? Is our thought axiomatic, or isn't it? It's a question of proof, not just of assertion. And this raises the real question of robots (i.e, machines that can mimic humans): if they acted like us, would they imitate our violence, too? That seems to be the basis of the robots in the new "Battlestar Galactica," and it's a fascinating idea. They have religion (ours; or at least a recognizably human one); and while they may think axiomatically (as a computer must; that's not what Turing established, by the way, but what his logic lead to: the use of axiomatic programming, algorithms, to make computer programming function), do they realize that's what they are doing? In fact, are they violent, or simply following their axioms? Since they can't "think" beyond them, to us their actions seem rather, well, one-sided. But are ours any less simplistic to, say, dolphins? Or to "god." For if there is an entity that does not think axiomatically at all, we would call it "god." At least, unless we can establish conclusively that we don't think axiomatically. (That quote from Wang below, BTW, citing Godel, is not "new age," but a fellow mathematician and close colleague in the later years of the logician's life).
A computer can't go beyond its axioms (essentially what Turing proved).
Ray Kurzweill famously described a thermostat as having some degree of consciousness. Doesn't pass the Potter Stewart test: I know consciousness when I see it, or, at least, I think I do, and my 120 lb fluffy white Pyrenees dog has consciousness, and a thermostat doesn't.
Goldstein makes the case well, I think, that Godel placed limits, not on human thought, but on axiomatic reasoning; that Godel felt not that he'd added to the uncertainty of the world but that he'd established a Platonist reality. Not the 'new age' (with due respect--no snark) take on Godel at all.
And no, Godel did not think he'd added to the uncertainty of the world; he was a devoted Platonist, but, as Goldstein is at pains to point out, the Platonism preceded the logic (Yourgrau, in a separate book, seems to think the reverse is true, but Goldstein makes a compelling case that Godel's theorem proceeded from his Platonism, and not the reverse). And I'm not sure he did, either; nor do I mean to imply he did. My interest is the limits of epistemology. In fact, my interest is in boundaries, altogether.
Yourgrau, in fact, uses a metaphor that is apt here: an "atom smasher for the mind." He writes:
In subatomic physics, one can submit particles to the extreme forces of an accelerator, or "atom smasher," which results in particles that are indistinguishable under less-extreme conditions revealing themselves as distinct. Godel would devise a method for subjecting the concept of space-time to similarly extreme conditions-in this case, geometrically, not dynamically, extreme-so that invisible differences between the two concepts would become manifest. This too was a continuation of a methodology G6del had employed in his incompleteness theorem. The method consists in creating what can be called limit cases, formal constructions that by design are so extreme that they limit, mathematically, the possible intuitive interpretations they will admit. For his incompleteness theorem, Godel devised a formal system together with a series of ingenious definitions and cooordinations for which it could be demonstrated that the concept of formal proof, as it appeared in the system, could not, on pain of contradiction, be interpreted as representing intuitive mathematical truth. He did this by constructing a formula that was provably unprovable, but intuitively true. Palle Yourgrau, A World Without Time (New York: Basic Books 2005, pp. 114-15)This is the level of proof I'm talking about. Not all questions, obviously, can be subjected to that level of scrutiny. But here again, we're up against an epistemolgical question: how do we know what we cannot know?
Classical theology (thanks to Luther) admits of a "negative theology," of that about God which we cannot know, or claim to know, or even speak of. It is an acceptance to the limits of our knowledge, and the basis for Kierkegaard's famous "offense," the "absolute paradox:" that eternal God could also become temporal, that that which is both the basis of existence and has no limits, could come into existence and be limited in time, subject to the limitations of mortality, which are of course beginning and end, birth and death. This we simply cannot comprehend. Does that mean it is beyond our abilities? Or that it is false? How do we determine?
One answer is to decide, or not be able to decide, the existence of God. But Kierkegaard meets us there, too. That, he says, is "the ultimate paradox of thought: to want to discover something that thought itself cannot think." Johannes Climacus (Soren Kierkegaard), Philosophical Fragments, tr. Howard H. Hong and Edna V. Hong (Princeteon University Press: Princeton, New Jersey 1985, p. 40) But does the inability to think it necessarily negate the existence of anything? No; nor does it establish it. As Johannes Climacus goes on to point out, we never reason in conclusion to existence, but rather in conclusion from existence. So any attempt to prove the existence of God presupposes the existence of God; and any attempt to disprove the existence of God presupposes the non-existence. Which makes both attempts tautological, in the end, and not really establishing anything.
And we're back up against the question of the limits of our knowledge. Unfairly, I admit; but all I can do here is poke the issues around, rather like stirring a fire to keep it going through the night. (And I never even got back to the question of consciousness!)