I'm going to take a stab at the ontological argument. It's the most intriguing and perplexing of the traditional theistic proofs. Plantinga, Gödel, and E. J. Lowe formulated important versions of the ontological argument, which merit serious consideration in their own right. But these are pretty technical. In this post I'm going to focus on a more Anselmian approach.
This isn't strictly an exposition of Anselm's argument. I'm not sure what he had in mind. Scholars differ on how to interpret his argument.
i) Moreover, anyone writing a millennium later will have conceptual resources at his disposal which were unavailable to Anselm. My primary objective in this post is to consider what kind of argument it is.
In addition, I will attempt to explicate a key principle of the argument: the internal relationship between what is and what is greater than (i.e. between existence and greatness).
ii) It seems to me that the ontological argument takes the form of a dilemma. Even an atheist must begin with a concept of God. Unless he has a concept of God, he can't deny God's existence. (Anselm's target is explicit atheism.)
The idea of God is either greater than the thinker who thinks it or not. If greater, then it must have a corresponding reality above and beyond the thinker. If we can entertain the idea of a greatest conceivable being, then that must be more than an idea.
Anselm's definition of a greatest possible being includes a being who cannot fail to exist.
iii) Anselm's barebones formulation suffers from a superficial equivocation. When he speaks of existing in the mind or existing in reality, he doesn't mean God in himself exists in the human mind. Rather, it's a type/token relation. Our idea of God is a token of the extramental reality. A God who exists in the mind is shorthand for a conceptual token of God.
iv) In Contact, the action builds to the climactic encounter with a superior alien intelligence. However, that climactic scene is anticlimactic. Indeed, it's bound to be anticlimactic.
That's because the alien is just a fictional character. The alien is not a superior intellect. Rather, the alien is just Carl Sagan's idea of a superior intellect. So it was inevitable that the big scene would be a letdown.
The alien character can't be any greater than its creator. It can't be wiser than Sagan. If it's just a human idea, then it can't surpass the source.
That's a problem for SF aliens generally. Since the alien is just a figment of human imagination, it can't rise any higher than the source.
v) Compare that to our notion of the Mandelbrot set. Mathematicians have a partial understanding of the Mandelbrot set. However, the Mandelbrot set vastly exceeds human comprehension. We're dipping our toes into a fathomless fiord. It's far too complex for the human mind to grasp in detail. We can only sample it.
In that case, we have an idea of something greater than ourselves. Something we discover. Because the Mandelbrot surpasses human comprehension, it must in some respect exist apart from finite human minds.
That's a type/token relation. Our idea of the Mandelbrot set is an instance of that greater reality.
vi) In math we also have unproven theorems and conjectures. That means a mathematician can be smart enough to think of a problem that's he's not smart enough to solve.
In principle, a mathematician can devise a provable (or falsifiable) theorem or conjecture which no mathematician will ever be able to prove (or disprove). These conjectures and theorems are true or false independent of whether we can solve them.
But in that respect the concept is greater than the thinker who conceived it. It isn't reducible to human cognition. It has a reality that transcends our efforts to mentally probe it.
vii) We could also relate this to truthmaker theory. Take counterfactuals. Must there be some corresponding reality that makes counterfactual claims true? What's the truth-maker for the truth-bearer? There's a relationship between what's true and what exists. Truth is contingent on being–of some sort.
viii) It seems to me that to succeed, an Anselmian-style argument must demonstrate that the idea of God is suitably analogous to (v) or (vi). That our concept of God is that kind of idea. Once established, that will, in turn, implicate the existence of God.
ix) A potential way to escape the force of the argument is to retreat into conceptualism or nominalism. Of course, that's only as good as the case for conceptualism or nominalism.