In a moment of uncharacteristic humility Richard Dawkins writes:
"...like...other fantasies that we can't disprove, we can say that God is very very improbable".(Why There Almost Certainly is No God, Huffington Post, 23 Oct 2006)
Where Dawkins goes wrong is in thinking that "God" is an empirical hypothesis which attempts to explain features of the world in the same way as physics and chemistry. Fundamentalists may indeed give this impression but their literalism is a 20th century invention reflecting the modernist worldview just as much as Dawkin's science and, as such, is a departure from Christian tradition. In Christian orthodoxy God is not an empirical explanation but a metaphysical account and, as every philosophy undergrduate knows, a metaphysical theory is consistent with every state of the world. So you can't settle the matter of God's existence simply by referring to observable facts.
However, I believe we can go further than Dawkins does and make the bold claim that it is indeed possible to prove that God does not exist. But it is not done by experiment or examination of facts but by logic and conceptual analysis. We can say that God is not just probably or actually non-existent—he is impossible and cannot (logically) exist!
I want to claim that in the same way square-circles, married-batchelors and the largest-possible-number are impossible (because they entail self-contradiction) the very concept of God contains so many internal contradictions that it is empty of content. As in the case of many other internally incoherent ideas this is not immediately evident. Take the notion of time travel as we see it depicted in H G Well's 'The Time Machine'. At a schematic level we can enjoy the idea of someone travelling backward and forward in time because we ignore the absurdity of treating time as if it had the same properties as space. When we analyse it closely we see how it falls apart as a notion.
The a-theistic argument that takes this approach is known as "The Argument from Incompatible Properties". It is quite ancient and can even be found in the Buddhist Pali scriptures.
It goes like this:
Names refer not to things as such but to descriptions (this enables us to refer to fictional entities). To ascribe contradictory properties to something is to both assert a description and take it back at the same time—and they just cancel each other out. The name is thereby rendered empty and meaningless with no reference.
Traditionally God is envisaged as an agent who creates the world, sustains it, guides it providentially, intervenes in it miraculously, reveals his nature and will through it and rolls up it up at the end of time applying justice to all moral beings. However God is also Perfect, Immutible and Eternal. It is therefore impossible for such a being to do anything. If he is perfect he will have no needs or wants and so have no motivation to create a world in the first place let alone act within it. If he is unchanging and eternal, action is impossible because acting is a process with states before and after the action. Without time and change action is impossible.
God's omnipresence and omniscience are incompatible with his personhood because being a person involves seeing the world from a specific point of view. An omnipresent and omniscient being would know every point of view simultaneously which is not to have any point of view at all. Again—acting from no point of view at all is incoherent
Transcence and omnipresence contradict each other in that a transcendent being is completely above and beyond the world while an omnipresent God is present in every part of it.
God is said to be perfectly just and perfectly merciful. In our impefect world justice and mercy are traded off against each other but a perfect being cannot trade off, he must exemplify both virtues completely. Justice means giving to a person what is due to them but when we treat someone mercifully we cede the right to give them what they deserve. You cannot treat the same person justly and mercifully at the same time.
These are just a few of the internal contradictions in the concept. Space prevents me from elaborating or dealing with counter-arguments. You might object that I'm being a bit too literal in my interpretation of language about God. Such use of words is common in fantasy fiction and poetry. Well, I'm happy to concede that. "God" is a poetic fiction. Indeed, I think that's the answer!