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Dawkins' central move (and his most egregious mistake)

He calls his central argument against God the 747 argument.  If God is doing all of the stuff that theists think, then God must be the supremely complicated being (like a Boeing 747).  But just as it is improbable that such a machine would come together spontaneously as the result of a tornado going through a scrap pile (a comparison he borrows),  so too it is supremely improbable that God's supremely complicated parts would come together without an antecedent designer.  God is the most improbable being of all.

He chronicles how he expressed this objection at a Templeton sponsored meeting and how he was reprimanded by theologians who, in his opinion, just didn't get it.

But it is a self-stultifying objection if there ever was one.  For the same arguments that support the existence of God also support the claim that God is simple.  It is clear from other passages in the text that Dawkins doesn't understand those arguments.  Or rather, his attempt at understanding borders on the comical.  So it is only natural that he would fail to see how those arguments render his objection moot.

And his other objection, closely related, that God too should be explainable, only sounds reasonable if one fails to consider the same arguments for God.  They demonstrate or attempt to demonstrate the existence of a Being that is first in many ways.  But those ways of being first by definition preclude the sort of scientific explanation RD has in mind. 

His objections amount to a kind of circular argument.  He starts by assuming that God, if existent, could only be a being with a body.  He reads the arguments for God in a manner that keeps that assumption front and center.  Such a reading keeps him from reading those arguments in an insightful manner.  This lack of insight keeps him from seeing how God, if existent, cannot be a material being.  Thinking of God as a super-person, a supremely complicated material being, he has the insight that such a being could not be first, as theists also suppose God to be.  He triumphantly proclaims that he has discovered a contradiction between God's attributes and s a complicated body or something analogous thereto.

***

More charitably, it is worth pointing out how Aquinas, when he talks about simplicity, sometimes points to analogies in nature. The sensus communis is simpler than the external senses, etc. 

If I were in a friendly conversation with RD, I would point to these analogies.

Using those analogies, I would form the following conditional (and its contrapositive)  If one denies  that God can be simple, then the basis for this denial may have implications for lessor instances of unity.  It implies that one act of a higher power cannot synthesize the acts of many lower powers.  Which implies that human beings do not possess then one denies the unity nature of human beings.  If one affirms the latter, however, then one must be open to the possibility of something analogous re God's inner simpicity.

It's that simple.
 

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