What do I believe is true even though I cannot prove it? This question has a double edge and needs two answers.
First, and most simply: "everything". On a strict Popperian reading, all the things I "know" are only propositions that I have not yet falsified. They are best estimates, hypotheses that, so far, make sense of all the data that I possess. I cannot prove that my parents were married on a certain day in a certain year, but I claim to "know" that date quite confidently. Sure, there are documents, but in fact in their case there are different documents that present two different dates, and I recall the story my mother told to explain that and I believe it, but I cannot "prove" that I am right. I also know Newton's Laws and indeed believe them, but I also now know their limitations and imprecisions and suspect that more surprises may lurk in the future.
But that's a generic answer and not much in the forward-looking and optimistic spirit that characterizes Edge. So let me propose this challenge to practitioners of my own historical craft. I believe that there are in principle better descriptions and explanations for the development and sequence of human affairs than human historians are capable of providing. We draw our data mainly from witnesses who share our scale of being, our mortality, and for that matter our viewpoint. And so we explain history in terms of human choices and the behavior of organized social units. The rise of Christianity or the Norman Conquest seem to us to be events we can explain and we explain them in human-scale terms. But it cannot be excluded or disproved that events can be better explained on a much larger time scale or a much smaller scale of behavior. An outright materialist could argue that all my acts, from the day of my birth, have been a determined result of genetics and environment. It was fashionable a generation ago to argue a Freudian grounding for Luther's revolt, but in principle it could as easily be true and, if we could know it, more persuasive to demonstrate that his acts were determined a the molecular and submolecular level.
The problem with such a notion is, of course, that we are very far from being able to outline such a theory, much less make it persuasive, much less make it something that another human being could comprehend. Understanding even one other person's life at such microscopic detail would take much more than one lifetime.
So what is to be done? Of course historians will constantly struggle to improve their techniques and tools. The advance of dendrochronology (dating wood by the tree rings, and consequently dating buildings and other artifacts far more accurately than ever before) can stand as one example of the way in which technological advance can tell us things we never knew before. But we will also continue to write and to read stories in the old style, because stories are the way human beings most naturally make sense of their world. An awareness of the powerful possibility of whole other orders of possible description and explanation, however, should at least teach us some humility and give us some thoughtful pause when we are tempted to insist too strongly on one version of history—the one we happen to be persuaded is true. Even a Popperian can see that this kind of intuition can have beneficial effect.