[editor's note: hopefully I've done myself and the topic more justice since writing what appears below.]
I don't mean that question the first way you're likely to read it -- I have no intention of passively surrendering to natural theology. Rather, the question put to me as a Barthian is, "Why not natural theology?" And I respect the man that put the question to me -- besides the fact that he's my boss -- because the question bears heavily on what he does for a living: religion and science. But at the same time, I wonder why he needs it. (And for all my respect, I'm going to go unceremoniously about my thought process because it's what this place is for.)
Barth's opposition to natural theology is quite well-known. He told the Gifford Lecture folks "no" repeatedly, and when they insisted, he did natural theology the best service he could: he presented the most solid case for theology based on divine self-revelation possible. He presented it based on the Scots Confession, in two parts: Gotteserkenntnis und Gottesdienst. The knowledge of God, and the service or worship of God. Why is this the best service he could do for natural theology? As he put it, natural theology is always a reaction against revealed theology. The strength of the one is directly related to the strength of the other.
And yet that's only the case for one kind of natural theology -- what I'm calling Protestant natural theology. I'm sure there's a more precise way of putting that, but the natural theology Barth reacts so strongly against is an Enlightenment product, and a break that is only conceivable after the Reformation. One break with the church paves the way for another. On the other side, for Catholics after Aquinas, the words "natural theology" have a different ring to them. They are utterly dependent upon revealed theology because Catholic natural theology describes God's revelation as it remains visible in the natural world, in our senses, etc., after the fall. While the Protestant variety can easily turn against God, the Catholic variety makes no sense without God.
He suggests a third option, philosophical natural theology, but I'm hard-put to see it as a genuine third. It is obviously not Catholic. And perhaps my language of "Protestant" suggests a sort of basic religious outlook, but the best of modern irreligious theology falls after the Reformation, pushing along behind it. It is what Barth opposes - knowledge of God from creation, known cleanly without God's giving Godself to be known. Knowledge of God that can be had, not only without the church's help, but without the consent of the Subject. Objective, naturwissenschaftliche knowledge of a subject that cannot evade detection, a subject naturally epistemologically available because nothing can escape being so. Hubris! (And besides, what a silly place to look for a deity! The only things you'll find there are things that disprove that there is a supernatural being bound within the systems of nature.)
What can be read from the book of nature is not equivalent to what is told by the witness of scripture. We learn a great deal about the ways the systems in which we are embedded function - and the ways they break. But without prior knowledge of the God to whom creation is ascribed, nature speaks no unambiguous word of God to the observer. God is not a component of the systems arrayed about us. As Tillich said, God is source and ground of the figure/structure - present around and within it, but not bound by it. As Barth said (III.1), creation exists separate from God, and this is God's intention, so that God and the creation should be in relationship. But as such, God is rightly hidden from our controlling grasp. Of all the things we can do with our God-given powers and gifts within creation, revealing God is not one of them.
Here is the trouble I see with natural theology: either it cheats, and knows the answer it wants in advance, or it makes a god of nature. (Or of the necessities thereof, the "fingerprints of the creator" ... but so often that, too is cheating.) By cheating, I mean many things -- there's no reason for the Christian God to be its desire, though in the Christian West that is often the model for the shape filled in by natural theology. But any of them are still cheating in the same way that any clean-room reverse-engineering process cheats: we have an objective, someone else's finished product that we desire, and we abstract the relevant attributes of that product because we can study them and know them in advance of our "discovery", and then implement them somewhere else. So deism, among many others. There is an operating assumption there that the truth of a religion can be abstracted from its particulars -- that it can be universalized in ways that allow one to evade the sticky details. The only honest natural theologies of my experience are aboriginal. (I'm not sure that's quite the word I want, but it'll do to convey the idea.)
The thing the Bible has going for it, is the simple fact that it is not self-evident. It cannot be derived from the natural world. Neither, therefore, can it be reduced to the natural world, even if the witnesses it preserves are witnesses to events that happened within creation. This is not to deny that, for the believer, all of creation declares the glory of God and shows itself to be God's handiwork. But it only does this for the believer, for the one to whom God has already been revealed in relationship to that one and all others. This is the result of faith understanding the world. And this changes in no way the basic functions of all the systems in which we are embedded within creation, except that we understand that God is Lord of them. This has always been something we ascribe after the fact -- the doctrine of creation is not derived from observation of the world, followed by a Newtonian insight that God must have made all of this stuff I see. (The Cartesian assertion that the creator is the second most basic logical element of the world, after the self, is no exception!) In the Bible, creation narratives are spread all across the historical development of the theology of the people of God, from the Psalms to the Deuteronomistic History to the intertestamental literature. But inevitably they are later developments, developments that follow the creation of the relationship of the people of God to God by God's great works for them. The people who live into this relationship grow in faith until they see and believe that all things are within the power of God, and declare by faith that all things come from God and return to God.
I'm sure there are responses to this position of mine, and things I haven't covered, and I'd love to hear them, so we can grow from the experience.