A critical test of anyone who appears to hold fundamental - or indeed fundamentalist - beliefs, is to ask the question, "What would it take to convince you that you're mistaken?" Someone who is convinced beyond reason that their beliefs are true will always reply to the effect that nothing would convince them. This is a sure sign of unreasonable fixed belief, and it's not worth anyone's time trying to argue them out of their position.
It's a question often asked of atheists, but Michael Shermer's flippant response, "A deposit of ten million dollars in a Swiss bank account under my name," doesn't count*.
For myself, the idea of a personal God who listens to your thoughts, answers your prayers, obsesses over what you do with your sexual organs and demands worship on pain of eternal banishment to some undefined unpleasantness, is absurd in the extreme and not worthy of consideration. I therefore find it hard to imagine anything that could convince me that such a God exists. Nevertheless, just because I can't imagine it, doesn't mean it's not a possibility, however remote. There's a chance I could be convinced of God's existence, but only if He convinced me Himself, in person, in a manner congruent with my reasonable standards for evidence. I will not, however, accept his petition from any third party.
The probability of such a petition is highly unlikely, I believe, but (as I've already indicated) not completely out of the question.
There is, however, an area where I would be more likely to accept the existence of some kind of intelligent creator - though this intelligence would bear little resemblance to the God of scripture.
If scientific analysis were to reveal that the Big Bang could not have been instigated by anything other than the creative will of some kind of intelligence, science might have to concede that the universe came about by other than natural processes. But speculation about what happened at, during, just before or just after the Big Bang appears to be mired in philosophy rather than science, with doubts about whether you can meaningfully say anything temporal or spacial about an event that brought both space and time into existence. (You might also question your definition of 'natural processes'.)
Similarly, if science were to show that the presence of information in DNA could not have originated naturally, the existence of some agency that inserted the information would have to be postulated. But we would have no reason to call that agent 'God'. Just because such an agent would likely be beyond our comprehension, we are not barred from speculating on its origins.
What applies to DNA also applies to the Big Bang. Calling the originator of the Big Bang 'God', or the originator of the information present in DNA 'God', simply stops all further speculation. It's an abdication of intellectual responsibility, not worthy of science, and should be deplored.
Science has yet to explain these things, but that doesn't mean the answer is "Goddidit."
There are many arguments for the existence of God, and the Argument from Design (also known as the teleological argument) is the least weak.
(*During a recent debate in Australia between John Lennox and Michael Shermer, the moderator asked Shermer, "What piece of evidence, formula, piece of reasoning, whatever ... what would cause you to believe in God?")
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