Just because I can't prove it, doesn't mean it's not true.
Or so goes the thrust of Tit For Tat's comment on DagoodS's excellent blog.
There are a lot of things wrong with this attitude.
First of all, as DagoodS notes, we can evaluate many claims — such as Tit For Tat's "par 4 hole-in-one" example — by appeal to circumstantial evidence. The disanalogy to claims about God is sharp: we lack even circumstantial evidence for claims about God.
Second, we have to look more deeply at "can't"; this is an equivocal term, because it can refer either to practical difficulty or theoretical impossibility, which are two very different issues.
The notion of time symmetry (reversibility) is extremely important: That physical laws are time-symmetric is another way of describing the conservation of mass-energy. This means that, in theory, we can make actual observations; if those observations are sufficiently accurate, we can "wind back the clock" using the time-symmetric laws of physics, and determine precisely what happened in the past. Even issues regarding the role of the observer in quantum mechanics do not prevent this sort of observation and retrodiction. Of course the practical difficulties of making such specific observations are overwhelming. But there is nothing that prohibits us in principle from doing so.
Again, the disanalogy to claims about God is sharp. The problems about knowing or proving claims about God are not practical, it's not like God is hiding behind my couch, or even around some distant star. The problems are theoretical: You cannot in principle know anything about a God who makes no observable difference in the world, at least not by empirical, scientific methods.
But the philosophical issues are most severe. What precisely do we mean by true but unprovable propositions?
For example, empirical science might not be the last word in epistemology, but if we could somehow know things about God by some alternative methodology, then the objection about the unprovability of true propositions would be rendered moot. Simply proposing an alternative methodology does not inform us as to the truth of propositions that cannot be resolved by any methodology. The claim is not, "Just because I can't scientifically prove it..." the claim is that "Just because I can't prove it at all..." The first version simply raises the question, "If you can't prove it scientifically, how can you prove it?" The second version raises more profound philosophical questions.
We intuitively believe that there are true propositions we don't know are true, therefore there propositions are true or false independent of our knowledge of their truth or falsity. But again, we must be very precise, because "independent of" has the same equivocal meaning as "can't" above. Do we mean independent of practical considerations, or do we mean independent of principled considerations, considerations that follow from the theoretical limits of our epistemology? None of our present knowledge was unavailable in principle to anyone, ever, and what we think we can't know we think we can't know because of the practical problems acquiring that knowledge. Under scientific methodology, ignorance is always a practical problem, not a theoretical problem. And the clear implication is that a statement that can't be be known in principle is not truth-apt.
Under a scientific methodology, truth is metaphysically subordinate to knowledge. To call a statement "true" is to just say that we know it's true, and to call a statement truth-apt is to know ("meta-know") that, in principle, we could know that it's true.
The problem with unknowable-in-principle "truth" is that by definition, because the truth is unknowable in principle, we cannot agree on its actual truth. By definition, no contradiction is entailed by believing the statement is true, nor is any contradiction entailed by believing the statement is false. Just entertaining the idea of unknowable-in-principle "truth" is to deny the essential semantic property of what we mean by "truth": true for everyone, always. Unknowable-in-principle "truth" is impossible to distinguish from opinion.
Of course, there's nothing wrong per se with opinions, desires, preferences, values, and other purely subjective entities. Having and acting on opinions is part of what it means to be human; we are not mere "knowing machines", concerned exclusively with knowing the truth.
But confusing our subjective preferences with actual objective truth is very dangerous, because we have very different ethical beliefs about truth than we have about opinions. "True" means "true for everyone"; if we confuse our opinions with truth, we must then conclude that someone holding the opposite opinion is mistaken, often dangerously so, in just the same sense that a person who believes that it is safe to drive while intoxicated is dangerously mistaken.