In recent days I've read two documents which, taken together, seem to have something important to say about human free will in the light of God's inner nature.
One of these documents is philosopher J.R. Lucas' 1961 article "Minds, Machines, and Gödel," the full text of which may be read here. I discussed that monograph in Whither the "Magic" of Mind?. The other is Pope Benedict XVI's September 2006 Regensburg lecture, which I talked about in The Pope of Reason. Its prepared text can be read here, or in PDF format here. The Wikipedia article on it and the controversy it ignited concerning the supposed difficulty of interreligious dialogue between Catholicism and Islam is here.
The real subject of the Pope's address at Regensburg was the "coherence within the universe of reason" that describes God's true inner nature, as well as that of his creation. Benedict affirms that "not to act 'with logos'" — the Greek term for both "reason" and "Word" — "is contrary to God's nature." In other words, God cannot do anything that is not in keeping with the fundamental coherence at the core of his very being.
Because God is internally coherent, he cannot say one thing and do another. He cannot pretend he wills this and actually will that. He cannot wear a mask for our benefit that belies who he really is. Nor can he create us in our world to be, at the deepest metaphysical level, anything but as rationally consistent and logically coherent as he himself is.
Lucas' "Mind, Machines, and Gödel" is likewise concerned with questions of internal consistency and logical coherence. Lucas' intent is to show that the universal "incompleteness" of formal systems that was proven by Austrian mathematician Kurt Gödel in 1931 means that no machine can duplicate the human mind.
A formal system is a system by which a large set of theorems can be proven, drawing on a small set of axioms and rules of inference. For example, a formal system can easily be constructed that will derive all the laws of simple arithmetic that we learn in elementary school.
Gödel took up the question of whether every formal system is intrinsically incomplete. "Incomplete" in this sense means "unable to prove all true theorems about itself." Gödel demonstrated quite rigorously that any system which was at least powerful enough to produce the laws of simple arithmetic was in fact incomplete.
That is, it was logically incomplete as long as the system was internally consistent. An internally inconsistent formal system is one that is able to derive perfectly good theorems that are mutually contradictory. In the most general formulation, "P is true" and "P is false" are the contradictory propositions that coexist within an inconsistent formal system.
In particular, Gödel constructed a proposition, call it G, whose English equivalent is "G [i.e., this proposition itself] is not provable within this system." That so-called Gödelian sentence is the Achilles' heel of any formal system. Gödel showed that, as long as a formal system is internally consistent and powerful enough to derive the laws of arithmetic, a Gödelian sentence G is unprovable within its own system. Yet we who stand mentally outside that formal system can see, by our knowledge of of G's very unprovability, that G is in fact true.
In his paper, Lucas extends Gödel's logic to show that our minds, which can indeed perform outside-the-lines feats of truth divination, could never be fully emulated by machines. Machines, Lucas says, are but "concrete instantiations" of formal systems, and they are as such equally limited by Gödelian incompleteness.
In discussions of Gödel's Incompleteness Theorem, as the key proposition proven by the Austrian mathematician in 1931 is called, there is an interesting bias. It is simply assumed that formal systems would rather be incomplete than inconsistent.
I find this interesting because the Pope in his Regensburg speech seems to be implying that God, too, prefers incompleteness to inconsistency. But God apparently prefers to both of these possibilities a third possibility: that we, his prized creatures, be free.
When the Holy Father asserts that God by his very nature creates and sustains a cosmos ruled by "coherence within the universe of reason," we are entitled to ask what might constitute the diametrically opposed notion. A God, we might suppose, who is prone to incoherence: an incoherence which equates to the sort of internal contradiction spoken of in the Lucas paper, as it pertains to formal systems.
Put another way, we as God's creatures might conceivably have been created as computer-like beings whose behavior is wholly determined by our programming (akin to a formal system's rules of inference) and our input data (akin to a formal system's axioms). We would then have been deterministic machines in a deterministic world.
True, we would have been unable to divine the truth of the likes of the Gödelian proposition G. We would, that is, not have been as smart or as capable of seeking and finding the truth as we actually are. But there would have been an even worse concomitant: we would lack free will. Deterministic machines are not able to follow paths of intentionality that violate their basic programming.
Of course, if said humanoid machines are capable of harboring logical inconsistency at their inner, most fundamental level, that would change things. Just as with formal reasoning systems, machines with any substantial degree of "processing power" — ones that are roughly as "smart" as we — can be either formally incomplete or logically inconsistent. If they are incomplete, they are blind to the truth of their own Gödelian sentence. Hence they are not as smart as we.
In fact, to be fully as smart as we — to be able to see the truth of Gödel's proposition G — a machine would have to be internally inconsistent, in terms of its ability to derive contradictory truths!
If Lucas is right, though, we are not machines at all. We could never be replaced by a machine, whether it is a formally incomplete one or a logically inconsistent one. We can conclude from this that God made us free in order to avoid having to build rational incoherence into us, at the very base of our being — given that his only other choice was to let us remain as mentally limited as a computer or robot that cannot see the truth of its own Gödelian sentence.
In other words, it seems that smart creatures that evolve in a fundamentally rational world created by a metaphysically coherent, non-capricious God must perforce have free will! Lacking free will, such creatures would have to be mentally deficient machines — either that, or the world in which they evolve would have to have been founded on the quicksand of a radical, through-the-looking-glass logical incoherence.
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