Leibniz reasoned to the existence of God by postulating that the universe might not have existed, therefore requires an explanation for its existence, and, unless you are going to argue that it's "turtles all the way down," there's ultimately going to have to be some self-explanatory cause. This line of reasoning certainly does not provide an airtight proof for God's existence... it rather seems to point us back to Swinburne's assertion that God is the simplest explanation. David Deutsch contributes a further insight: "important explanatory advances often change the meaning of explanation." For example, we might conclude that the part of reality we can see is not representative of the whole, like those prisoners in Plato's cave I mentioned before, in which case, unless we can either get out of the cave or receive and understand information from outside it, our "explanations" are hardly worthy of the name.
Jim Holt introduces another important concept, related to the self-explanatory explanation, about halfway through the book: that of the self-subsuming principle. Robert Nozick, known primarily as a political philosopher, developed this concept. A self-subsuming principle is sort of the opposite of the paradox where a man comes up to you and says, "Watch out! Everyone in this town is a liar!" What are you supposed to do with that information, when it contains its own contradiction? On the other hand, if he said, "Good news! Everyone in this town tells the truth!" you would at least be free to believe his statement without cognitive dissonance. However, while that concept means that we can imagine explanations for the existence of the universe that are coherent, it gives us no way to be sure they are true. For example, if it is true that anything that can happen, will happen, then that statement happened, and everything has happened, including both nothing and all variations on the theme of nothing. Or maybe that's not how it went.
Einstein said the universe was eternal, and infamously adjusted his equations in light of that assumption. However, with the Big Bang looking more and more likely, scientists are speculating that quantum instability might have caused the Big Bang. After all, "changeless emptiness is incompatible with quantum mechanics." (p.141) Apparently, however, for reasons I still don't understand, this model is incompatible with gravity, thus necessitating the Theory of Everything that people are always on about. Furthermore, even if the conflict between quantum mechanics and relativity could be resolved, there's the question of where those two forces came from. If the universe is an unsurprising, indeed a predictable result of these laws, are they a transcendent reality? "Since the world is logically prior to the patterns within it, those internal patterns can't be called on to explain the existence of the world."
Now do you see why Swinburne finds the God Hypothesis the simplest, the most elegant, the most self-explanatory explanation for the existence of the universe?! The mind of God would be a great place for the laws of physics, quantum and relative, to reside, and His will would be a great catalyst for them to act.
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