When I refer to a "worldview" I am referring to a person's basic or foundational assumptions about life/reality. I suppose it doesn't necessarily have to be termed "worldview." You could also call it one's "basic ideology" or "theory of reality" or "philosophy of life." A worldview is a network of presuppositions concerning metaphysics (what is real? what is possible? what is existence?), ethics (what is right? how do we know what is right or wrong?), and epistemology (what is knowledge? how do we know? how is truth determined?). Related to this are a person's beliefs about where we came from, who we are, what our purpose is, and where we are headed. A worldview is a set of assumptions or presuppositions by which we reason and interpret our experiences.
Some people haven't given a whole of thought to their worldview or tried much to articulate it; some have. Some don't even care much to think about it; some do. But everyone operates according to a worldview. For example, if someone says something like, "Stealing is wrong," the reasons they give for upholding that statement begin to reveal something about their worldview as is pertains to ethics. Or if I say something like, "A truth proposition has the opposite value from its negation," my explanation for believing that begins to reveal something about my worldview as is pertains to epistemology.
A person's worldview is most poignantly revealed when the issue of God's existence arises. This is because one's belief about God is at the centre ("center" for my American friends) of one's worldview. In fact, what a person believes about God conditions his beliefs about metaphysics, ethics, and epistemology. One's beliefs about God have a significant bearing upon what one's worldview will be. This is the case for the Christian theist like myself and for the atheist. But when the theist and the atheist debate the existence of God, they do so in accordance with their worldviews. Both regiment their worldview when they reason about and debate God. This means, therefore, that the debate over God's existence isn't just a debate over isolated claims (e.g. God exists, Jesus rose from the dead), rather, it is actually a clash of entire worldviews.
Recall what I just said, that our worldviews are determined by what we believe about God. And yet, we use our worldviews as a basis on which to reason about everything, including God's existence. Both the atheist and the theist can be found reasoning in a circle. What the theist deems acceptable evidence and argumentation for God's existence will not always be found compelling by atheist because the atheist operates according a different epsitemology informed by his rejection of God. The only way to solve the dispute between the theist and the atheist is to place the one worldview beside the other and reason about them on their own terms to see which one can provide the necessary outlook on life to know or prove anything whatsoever.
The problem for the non-Christian, then, is that if you're going to reject God, if you're going to write off the Bible and Jesus Christ as superstition, can you come up with a worldview that makes sense out of reality? Can you account for right and wrong, truth and falsehood, possible and impossible, given your rejection of the God who created you and calls you to repentance for your sin? The problem for the atheist is that, if he's consistent with his atheism, it's not possible to come up with a workable worldview that offers a foundation for knowing or proving anything.
The reason that the atheist has successes in acquiring knowledge, making predictions, and accomplishing various tasks is only because reality is what God created it to be. The extent to which the atheist has any successes in his reasoning is only to the extent that he assumes the truth of the Christian worldview. In fact, even the attempt to prove atheism assumes the truth of theism. Both the attempt to prove and disprove Christian theism proves Christian theism because of the impossibility of the contrary. The atheist can't come up with a consistent, workable worldview. They can't account for the preconditions of intelligibility that make proving or knowing anything possible, in which case they can't prove atheism, and even the attempt to prove atheism ends up proving the truth of the Christian worldview.
(Wow, I've used alot of space and I haven't even replied to your comments about science or logic yet. But I'm more than willing to discuss these things if you want. In fact I'd like that because I still don't think you gave an adequate answer to the last paragraph in the previous post about gaps in knowledge. I'll even buy you the next cup of coffee, so to speak. Or perhaps a beer. In the meantime, if you want to learn more about how the laws of logic require God's existence, see this post here.)