Because of what Romans 1:19-21 says, I always thought that every person was given the capacity by God to know He exists without having to have been told by somebody.I think that's exactly correct. But it also illustrates a point that David Craft mentioned while quoting Simon and Garfunkle: "a man hears what he wants to hear and disregards the rest."
Helen Keller seems to have been a perfert example of this. After she gained the ability to communicate with the world she lived in, when asked if she knew about God, she replied that yes she did know Him but did not know His name.
By scripture it would seem that no one will stand before God and claim ignorance of Him as justification for their sin, they will be without excuse.
Belief, as much as we would like to think it's solely an act of the intellect ("show me evidence and I'll believe") is shown by both scripture and experience to be primarily an act of the will. That's one reason Jesus refused to trick-monkey his miracles. When people told him, "show us and we'll believe," he consistently said that belief was not merely a matter of being shown, but required willfil effort on their part.
Whether in matters divine or mundane, humanity has an incredible propensity to find evidence that supports what we choose to believe. Why is the wife the last to know? Why can some people honestly believe in unseen, worldwide, centuries-long conspiracies or aliens or thetans or virgin births? Because belief is a choice.
But these choices are especially troublesome for intelligent people (who tend to lean on the intellect) who also have a moral reason to deny the consequent, because they forget, in their sometimes honest attempt to justify unbelief, that they are just as prone to self-deceit as other men. Perhaps they are even more prone. And yet according to God, such unbelief, chosen as a matter of will rather than evidence, still has moral consequences.
That's why Jesus said, "You must (moral imperative) believe." There were plenty who saw his miracles and refused, even plotting to murder Lazarus after Jesus raised him from the dead. There are plenty who believe without evidence many things, including but not limited to belief in God, simply because they choose to do so entirely for reasons of their own.
Does that make evidence, facts, or science worthless? Of course not. For most facts there is no moral consequence. If I believe the earth revolves around the sun or the sun around the earth, neither affects whether I ought to bed my neighbor's daughter. That leaves such facts free to help us, to be applied widely, to improve our lives.
But there are certain beliefs that have such consequences. If God exists, then it's quite possible that he cares very much whether I bed my neighbor's daughter. And if he has some moral authority to say that I oughtn't, then I have two choices: I either obey him or I get rid of him. And I have built into me the ability to choose either and bear myself the moral consequence of such a choice. It is no surprise that the 'science' over which the most rabid fighting occurs (and those areas most wrought with fraud) are those that directly impact the above question.
I think the reason the existence of God cannot be "proven" is simply because He calls us to choose to believe. If we look at the beauty of the stars, we see power and majesty. If we look at a lion eating a gazelle or a young child dying of plague, we see futility, cruelty, and meaningless. "Vanity of vanities," saith the preacher. "All is vanity and a striving after wind."
And therefore each must choose. And what we choose we are confirmed in, by God and our own will, as Paul says in the passage Honorbound_Men quoted in the comments.
But whatever we choose, that we are responsible for. TANSTAAFL. Opportunity cost. Every rose has its thorn. Unless you believe that I am He, you will die in your sins.
For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, even his eternal power and Godhead; so that they are without excuse.
Either God rules or we rule. Either He is the final measure or we are. The choice is ours, either through delegation or because we are the highest intelligence on the planet. And that choice often decides the rest of the choices we will make in life, wise or unwise, for good or evil.