Penrose and his book have been debunked. http://www.mth.kcl.ac.uk/~llandau/Homepage/Math/penrose.html
The hostility the Strong AI camp have for Penrose's views is fascinating - it must be infuriating to have such a respected mathematician and physicist take the time to write a few books refuting the reductionist approach. There certainly seems to be no room for a contrarian around those parts!
Thanks for the link, but to state that it debunked anything does not seem to be correct. It was an attempted refutation by a computationalist, and Penrose answers these criticims in: http://web.archive.org/web/20080618195657/http://psyche.csse...
I'd recommend reading the section "4. The "Bare" Gödelian Case". Two particularly relevant points!
"4.5 The many arguments that computationalists and other people have presented for wriggling around Gödel's original argument have become known to me only comparatively recently: perhaps we act and perceive according to an unknowable algorithm; perhaps our mathematical understanding is intrinsically unsound; perhaps we could know the algorithms according to which we understand mathematics, but are incapable of knowing the actual roles that these algorithms play. All right, these are logical possibilities. But are they really plausible explanations?
4.6 For those who are wedded to computationalism, explanations of this nature may indeed seem plausible. But why should we be wedded to computationalism? I do not know why so many people seem to be. Yet, some apparently hold to such a view with almost religious fervour. (Indeed, they may often resort to unreasonable rudeness when they feel this position to be threatened!) Perhaps computationalism can indeed explain the facts of human mentality - but perhaps it cannot. It is a matter for dispassionate discussion, and certainly not for abuse! "
Godel's Incompleteness Theorem (in essence) said that the first order Peano axioms for the integers was not strong enough to prove all statement in the second order Peano axioms. My understanding is that mathematicians wanted a computable system that would be strong enough to prove all true statements encompassed by the second order system.
Godel showed this can't be done. No computable system can be strong enough to prove all true statements in the second order system. Given that humans can prove statements that are true in a system that can't be reducible to a computable set of axioms how can computer intelligence ever equal human intelligence? Human intelligence must be fundamentally not a computable system. What's wrong with this reasoning?
The true but unprovable statement referred to by the theorem is often referred to as “the Gödel sentence” for the theory. It is not unique; there are infinitely many statements in the language of the theory that share the property of being true but unprovable.
Basically, what you need to prove to show that humans are intrinsically more powerful than machines is that we have the ability to generally solve noncomputable problems in a general way, ie, that we are hypercomputers.
So the first order axioms were invented. Godel showed that the first order axioms are not strong enough to prove all true statements about the integers. There are infinitely distinct models of the first order integers. There is only one model of the second order integers.
In the proof of the Incompleteness Theorem Godel proves a statement about an integer which is not provable in the first order system.
Humans can work in the second order system but not computers. So, from my perspective there is something different about human thought patterns. I'm not an expert and would like to know what is wrong with my reasoning.
First Theorem: "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory."
Second Theorem: "For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent."
It wasn't that Godel proved that Peano arithmetic was incomplete that shocked mathematicians, it was that he showed that _no formal system_ could ever be complete as long as remained consistent.
Of course maybe this second order system isn't consistent, in which case there's no problem. After all, I'm fairly convinced that human reason isn't always perfectly consistent, so it makes sense that we can avoid Godel's theorems. Of course, I see no reason we couldn't make a slightly inconsistent computer either.
One can find complete axiomatic systems of the integers. It's quite easy. Just take all true statements as the axiom system. The problem is that there is no computable method for determining in such a system whether or not a given statement is an axiom.
Again, we humans can work in the second order axiom system but computers can't. So it appears there's a difference between human intelligence and computer intelligence.