This question has been debated over sufficient millennia that one might suppose there was no new ground left to cover. And if truth be told I haven't done the homework to ensure my contributions are novel: I am not trying to do scholarship, only self-entertainment. However I do have two points to make. One, on computational theology, will be the subject of another post. This one will deal with the following question: does the question "does God exist" even mean anything? Before we can decide on the truth or falsity of a proposition, the proposition must have a meaning. The theists say yes, the atheists say no; here I will explore a third option: the question is meaningless.
(Amendment: I couldn't resist checking, and Google and Wikipedia have made scholarship too easy even for a would-be lazy scholar such as myself: the position I espouse here apparently is not not novel, and has a name, Ignosticism.)
For simplicity let's restate the question as an assertion "God exists" to which we wish to attach a truth value, or at least evidence pro and con. With only 2 words to make sense of, it should be a straightforward exercise to determine whether the the assertion is meaningful. We just need to know what we mean by God, and what we mean by existence.
Let's start with God. Clearly this term comes with some baggage which contributes to an appearance of meaning. Big white guy with beard, robes, sandals. Invisible. Old, ageless immortal and/or eternal. Creator of the universe. Omnipotent, except for an inexplicable self-imposed reluctance to interfere with the exercise of human free will.
Some of these attributes are debatable, even among theists, as to whether they are truly attributes of God. For example, not all believers subscribe to God's whiteness, maleness, singularity (as opposed to plurality -- a "he", or maybe a "she", perhaps even an "it", but not a "they"). Many would agree that God would still be God if he turned out to be a she, or not white, etc. This suggests there are attributes of God that are inessential -- God may or may not possess them, and it is a question of fact whether these descriptors really are true of God. However if I said that my dog Norman, a perfectly respectable Springer Spaniel, is God, most believers would regard the assertion as untenable -- there are necessary attributes of divinity that Normal simply does not possess. (On the other hand if I say he was a carpenter in Gallilee circa AD 0, the question becomes less clear to some.) What are these necessary attributes of divinity such that without them a god candidate cannot be God, as opposed to those other qualities about which believers can debate without calling into question the idea that in using the term "God" they are referring to the same thing?
If we pick a neutral term, such as Bleurgh, and ask whether Bleurgh exists, the question is clearly meaningless. You will ask, rightly, what I mean by Bleurgh. Clearly therefore, for "God exists" to mean more than "Bleurgh exists", some of these attributes that form the penumbra of connotation around the word "God" must be sufficient to amount to a definition, i.e., necessary and sufficient conditions for Godhood. To put it another way, how much would I have to tell you about Bleurgh before you could determine that Bleurgh is God.
For many believers the sine qua non of Godness is Creatorhood: whatever else he/she/it/they may or may not be, God created the Universe. Nowadays we have a pretty good idea of what the Universe consists of -- lots of matter, energy, dark matter, black holes -- as a well as a God-awful amount of empty space, truly cosmic distances. So, good, we have part of a definition. However, suppose the next bulletin from CERN informs us that the Higgs boson in some incomprehensible mathematical sense created the universe. Does that make the Higgs boson God? Despite the clever marketing inherent in the catchy term "God particle", the equation of the Higgs boson with God is pretty much a nonstarter for theists. Clearly there is more to Godness than creatorness. Inanimate forces don't cut it.
An essential aspect of theism, at least in its Western Judeo-Christian incarnations, is that God is in some sense a person. This is a very peculiar belief, on the face of it, since no persons I know go around creating universes (except perhaps J.R.R. Tolkien and G.R.R.Martin, and they borrowed heavily from this one). Anything that goes around creating universes would seem to be such a drastically different sort of entity from ourselves that it is hard to see what "personhood" could possibly even mean as a category that includes both ourselves and her/him/it/them. Does she/he/it/they (I don't dare suggest an acronym for this array of potential pronouns-- maybe I'll stick with "it") have a biographical sense of her/his/its/their own life as a story with a beginning, a middle and perhaps even an end, which is so central to our own personhood? If immortal, eternal, uncreated, without beginning or end, it is hard to see how it could. Christianity and Hinduism, by positing incarnated Gods who intersect with human history, or Judaism, with its miraculous interventions, perhaps are attempts to span this gulf between two such drastically different types of persons. Christianity, in particular, identifies the creator of the universe with a Galillean carpenter, and considers it rude or heretical (punishable by death for most of its history) to point out the resulting cognitive dissonance.
Hindu concepts such as Brahman or Atman are so abstract as to make the question of personhood, or for that matter existence, beside the point. Their definitions are almost anti-definitions, in that they do not admit a yes / no answer to the question "is Bleurgh Brahman?" (or Atman) regardless of how Bleurgh is defined.
So our investigation so far as led to the following conclusion: for "Bleurgh exists" to mean the same thing as "God exists", we would have to intepret "Bleurgh" to be some sort of person (or persons) who created this universe, and we understand the cultural scope of this statement to be limited to Western traditions of a personal God. With those restrictions we might yet be able to make sense of "God exists" -- though we would still have to specify in more detail what personhood consists of, to avoid the Higgs-boson-is-God option. (Is the Higgs boson a person? Why not?)
Even if we can solve that problem, however, we still have to determine what we mean by "exists". There are different types of existence. One is temporal: things exist for a while, then they stop existing. For example, George Herman Ruth, aka the Babe, existed during a certain span of time. Did he exist 3 x 10e-37 seconds after the Big Bang? No. Does he exist now? Not really -- except as a mouldering corpse, or an impression on a variety of historical records and memories, or perhaps as an immortal soul in Heaven -- but not in the sense in which he existed between February 6, 1895 and August 16, 1948, when we was alive. So one notion of existing is a fluent: something exists at some times and not others. For most things, there is a single interval of existence preceded and followed by nonexistence, though some things come and go, such as the Black Death, the Crusades. the Wars of the Roses, el Nino, and so on. These recurrent events can always be decomposed into sets of simple events.
Another case: does Tyrannosaurus rex exist? First answer: not any more: it did once, but now it's extinct, its fluent has turned false. Second answer: there never was a single entity called T.rex; rather there were presumably a series of individuals who might qualify for membership in the abstract class of T.rexes. (Some issues there...) A more complex example: does (or did) brontosaurus exist? This may seem analogous to the T.rex case, except that paleontologists have decided that brontosaurus never existed at all!
What does this have to do with God? Well, if God existed in the sense in which other historical objects exist, then (ignoring the possibility of intermittent existence) we could ask whether God's (alleged) existence was possibly a mistake, like brontosaurus, or else had a start and a (possibly future) end date. Maybe God existed once but exists no longer -- God is dead? Maybe God died giving birth to the universe (and was therefore presumably female)? God's beginning must precede the universe's given his/her/etc. role as creator, but that is problematic from the standpoint of modern cosmology, in which time's existence and that of the cosmos are inextricably linked -- how could God come before the Big Bang if there is no before the Big Bang?
Thus a God who exists in a historical sense is conceptually problematic. What other options for existence are there?
Mathematics offers one. For every integer there exists an integer one larger. For any two distinct points on a line there exists a point in between them. There exists a number that whose square is 2. There exist 5 3-dimensional solids with regular polygonal faces.
Some theologians have tried to use mathematical arguments to prove the existence of God, much as one might proof the existence of the square root of 2. The results have not been convincing to believers or mathematicians.
Mathematical existence has some appeal as a home for Gods. It is eternal, timeless, independent of our particular universe. However the Platonic world of mathematical objects seems incompatible with God's aforementioned personhood. It may be that any locale that satisfies the need to place God outside of time, so as to be present before the Creation and so bring it about, is incompatible with the dynamic requirements of personhood, namely, to behave. This conflict may not be insuperable: the universe, including all of its history, may perhaps be viewed from without as a static object, with a spacelike all-at-once God's eye view of time replacing our in-the-moment view. In that case God might be able to be present in time while simultaneously existing outside of it, much as the Sphere in Edwin Abbot's Flatland could exist in the plane and beyond it at the same time -- and could do things that appeared miraculous to the denizens of Flatland.
Though there is ongoing debate as to whether mathematics is created or discovered, it seems clear that aspects of mathematics lie outside mathematicians' creative control. Mathematicians are not free, for example, to declare pi equal to 3, or to declare 9 to be prime. Would this impose a limit on God's creative control as well -- is God free to declare 9 to be prime? Is God subservient to rationality or does it get to decide the digits of pi as well as the structure of the universe?
A third alternative is to exist in the sense that "other universes" can be said to exist, in the fashionable physics concept of the multiverse. These other universes come in two potential flavors: ones that can't interact with ours, and ones that can. The former are interesting to speculate about, and can explain aspects of our own by Anthropic arguments, but cannot influence our history. The latter, exemplified by the notion of colliding branes in a multiverse, permit a degree of influence to leak between universes. God resembles these interacting branes, existing outside of our universe yet influencing it, and possibly creating it. Of course the multiverse as presently understood seems no better a home for personhood than the Platonic world. Maybe they are the same!
So having rambled to the end of my inquiry, is the question "does God exist?" even possibly meaningful? I suppose I conclude that at least one coherent meaning is possible: whether a personified creator of universes exists in either a multiversal and/or Platonic context, subject to further refinement of the definition of personhood. Gotta say, though, seems unlikely...