If we assume that a God could possibly exist then there are three basic options for how God and the world interrelate. The first is called Pantheism. Pantheism basically says that God and the world are one and the same. God is everything that exists and everything that exists is part of God. If we were to picture God and the world as two circles drawn on a piece of paper, then pantheism would have both circles completely overlapping. This is what I call the Star Wars God, he's like the Force, permeating everything. But that's not the kind of God I believe in.
The second option is called Deism. Deism sees the two circles of God and the world as completely separate, with no overlap at all. The realm of God's activity (i.e the "supernatural") is completely different than the natural realm and vice versa. Some deists say that God doesn't interact or interfere with the natural world at all, while some say that if he does interact it is always as an outside force "breaking in" in what we would call miraculous (i.e. unnatural) ways, but that these interventions are rare and unpredictable. We could picture this kind of divine intervention as as an arrow going from the God circle to the world circle, though perhaps with a dotted line to indicate their infrequency.
Most of the criticisms I hear from atheists about the lack of evidence for God's existence seem to be assuming a deist God. Everyone seems to be looking for evidence of some outside supernatural force breaking into our world in some miraculous, extraordinary way. Of course, it makes sense to assume this since most Christians even, at least since the Enlightenment, are essentially deists. We have slipped into this mindset of thinking about God and nature as operating in two completely different spheres of existence, so that God is reduced to being the "God of the gaps", i.e. whatever phenomenon science can't yet explain is where we insert God. But personally I think this is a very poor way of thinking about God.
The third option, and the one I hold to, could, IMO, be called biblical theism. This view sees God and world not as merely identical, nor as completely separate, but as distinct but intricately intertwined, with many points of overlap. This world is God's creation, and so we see his presence in it. The emotions we feel when we marvel at the beauty and wonder of the natural world are just what we would expect to feel if we ourselves were made in the image of an artistic God who delights in the beauty and wonder of his own creations. The love that we experience in this life points us towards the belief that love is actually the foundational fact of the entire universe (for God is love). The feelings of spiritual connection, ecstasy and transcendence that people from every culture and religion (even some atheists) report having experienced at various times, could be a evidence that there is a spiritual dimension to our lives that is interwined, not separate from our natural existence (after all, I most often have those experiences in some of the most ordinary of circumstances - when I'm sitting on a hillside watching the sunset, or chatting around a campfire with close friends). And our experiences of injustice and brokenesss and our universal longings for the world to one day be made whole and right again is a hint that maybe there really is a "way things were meant to be" in an objective sense and not just in our utopian fantasies. Those hints and evidences are those points in our experience where God's realm and the natural realm intersect and overlap. They are what the Celtic mystics would have called "thin places". In fact a Celtic knot is probably the best way to represent this view of God and the world. Two distinct circles, but constantly overlapping and interwining to the point where you can hardly separate the one from the other.
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