There was a lot of talk about “proof” in the news this month. The Ken Hamm/Bill Nye debate captured a lot of attention and raised questions about what science can prove about particular creation models, and even about what kinds of evidence are acceptable or reliable. There was an interesting story about a computer-generated math proof that required 13GB of reasoning, far too much for human beings to parse through and verify, which led to questions about the limits of what can be proved in mathematics and whether math will ultimately become the domain of computers. Meanwhile, an archaeological paper about camels led to a lot of headlines about the Bible being contradicted or perhaps even proven false.
All of this stirred up some of my own thoughts on what can (or cannot) be proved about God. For many Christians, it seems desirable to have logic or philosophy or science prove that God can or must exist. And yet empirically, it would seem that one consequence of pursuing such proofs is that we raise the possibility that God is potentially falsifiable. So naturally some folks become interested in proving God can’t or doesn’t exist, which seems to result in the perception that findings like the timing of when camels arrived in Palestine can falsify the Bible, and possibly by extension God. If the outcome of our attempts to prove that God can or must exist is to cause people to misunderstand the nature of the Bible or the nature of God, then perhaps we need to reconsider the claims we make. For instance, what if we instead tried to prove that God’s existence is somehow formally undecidable?
What would such a proof look like? The following is more of a sketch or an outline of a proof, or some musings that could perhaps one day be formalized; they are the thoughts that I’ve had on the topic and I’d be curious what other folks make of them.
First, let’s consider what axioms we can choose. We could start by assuming there is no God, in which case there is nothing to prove. Likewise for the assumption that there is a God. So the only interesting choice is if we assume that God is a possibility, or choose a set of assumptions that don’t explicitly say anything about God one way or the other. I suspect there are a range of ways to formalize that idea, but I think the notion of God as a possibility is sufficient for our purposes.
Now, we need to know what we mean by God. One way we describe God is that He is infinite. Here, I take that to mean that it would require an infinite amount of information to accurately describe God and His works. (This is admittedly a potential point of departure for some, but we need some formalization of what we are talking about and this one makes sense to me.)
Then let’s say we have some proofs that God does not exist. Let’s put them in some kind of order, maybe the order in which they are proposed. Each of these proofs must define the God they are disproving, and information theory says we can express that definition of God as a sequence of 0s and 1s (or bits). So now we have a big list of such sequences, each of which disproves a particular definition of God. But now we go down the list, and if the first bit of the first sequence is a 0, we choose a 1, and if the second bit of the second sequence is a 1, we choose a 0, and so on. After we are all done, we will have a description of God which is different than all of the other sequences, and thus it is a definition of God that has not been disproven. This suggests to me that one cannot disprove the existence of an infinite God.
But just because we cannot disprove the existence of an infinite God doesn’t mean we can prove it. In a finite amount of time, our definition of God that we are proving can only be finite, which means that the best we can do is prove a finite approximation of an infinite God, which, since the approximation is finite, does not really prove anything about an infinite God.
The combination of these two lines of reasoning at least suggest to me that an infinite God is in fact an undecidable proposition for finite beings, rather than being provable one way or the other.
Now, what are the implications of this conclusion? Well, an interesting consequence is that the probability of us finding the correct (finite, approximate) description of an infinite God among the infinite possible descriptions is vanishingly small. That might seem disheartening, but it implies that if such a God wishes to be known by finite beings, then He will have to initiate that process. And this is exactly what the Bible teaches — that we would never know God if He had not reached out to us. A provable God would have no need to do so, because we’d eventually work out the proof. And a God that provably does not exist wouldn’t exist to reach out to us. And so it would seem to me that the story of scripture is most consistent with a God whose existence is formally undecidable.