Given the troubles expounded by Hume on inductive reasoning, it's fairly well-established that proving things is impossible. But I take your point. There are, and have been, over the ages, many events that can be taken to be the hand of God meddling in human affairs, but can also be taken, by those who don't believe in a god, as mere coincidence, or chance, such as unexplained, spontaneous remissions. Granted, these are
not absolute proof that a god exists, but I feel that they give at least some credence to the belief in a god.
Just because a lot of people believe in a god(or a god, gods), doesn't make them correct. One simply cannot believe something into existance.
I'd also like to explore this idea. If something's existance or non-existanc makes no difference to the world beyond how people act, based on their belief in this thing's existance, can believing in it make it exist?
For example, say, somehow, it were proved, mathematically that God either did or did not exist. The physical world would remain the same, given that god's existance or non-existance had been the case all along, however, human interactions would change drastically, simply based on their knowledge of this fact. It seems to me that, in this case, simply believing in a god is analogous to his existance.