A slightly mismatched pair of arguments -- with one Anthony Flew disbelieved, and with the other he believed. Both are about science, both about God, and both, I think, are ideas that fall apart.
The philosopher decided as a boy to disbelieve in God because of unfalisifiability, according to his obit in the New York Times, but then, as an old man, he changed his mind and chose to believe1 because of unexplainability.
The latter argument seems to be particularly bad.2 For one thing, it equates God with whatever it is we don't know, an algebraic x, an answer that isn't an answer but a placeholder for whatever the answer will be. Answers do come, too, so the scientific unknown shifts, over time. Thus the belief in a God who is the action supplying the solution to problems of physics or biology is either belief in an ever-shifting thing -- what was once God is now gravity, what is now God might tomorrow be a newly-discovered particle -- or, more perniciously, a belief that requires one to hold some problems are unsolvable and the abracadabra! premise, which is not a solution but a question or blank space, an unknown, is to be considered enough, and no other solution should be sought. This is a bit like proposing 2 -x = 5 as the answer to 2 - x = 5, giving up the whole project of solving for x.
The former argument is more interesting and raises a good question: what evidence would make one reject the idea of the existence of God?3
I wonder, though, if it doesn't mistake God for a philosophic or scientific theory, and stretch those standards of proof further than they can really rightly be stretched. It doesn't seem, for example, like there could be any falsification test for hope or love -- what evidence would make you reject the idea of the existence of love? Not that God is a necessarily analogous to hope or love, but certainly the world as we understand it involves our acceptance of some non-scientific, unfalsifiable things.
It's interesting how closely related and yet asymmetrical these arguments are. In the obit they bookend his life. Both of them, though, seem flawed.
1. The God rejected and accepted were not the same, however. He rejected the God of Methodists, and, as an old man, confessed to believing in something like Aristotle's God.
2. There is a different and, I think, stronger argument to be made for the Aristotlean God, in which that God operates as a logical premise, but not a scientific one. It's stronger, in part, because logic is fixed in a way science isn't.
3. This is a profoundly worthwhile question in that it clarifies what is being taken to be God, and also in that it reveals the ground of God, which then raises the very fideist/Karl Barth/Kierkegaardian question of whether that ground isn't given a higher authority than God. For example, if you believe in Jesus on the basis of history, then don't you believe in history in a more fundamental way than you believe in Jesus?