I ran across something yesterday that disputed the validity of Anselm's ontological argument using the original (and long refuted) objection that says "I can conceive of a <insert category/type of thing here> than which no greater can be thought, but that doesn't mean it exists." This was originally proposed as an island by one of Anselm's contemporaries, Gaunilo, and Anselm swiftly refuted it.
When I first learned the argument in college, and the objections, I have to say that it seemed then and still seems now to be sound to me. (BTW, this is one of the best plain-language explanations of it I've found.) That said, I have always felt that the argument lacks compelling force. On the one hand, it seems cogent, and I don't see anything wrong with it in itself; on the other hand, it has a feel of verbal/logical trickery to it. You're left feeling like it makes sense and yet that it doesn't. It's the same sense you have when you see a magic trick, and you know that it's not magic, but you can't figure out how it isn't magic. Bertrand Russell seemed to agree with this, saying that it's much easier to be persuaded that ontological arguments are no good than to say exactly what's wrong with them. I think the reason it feels wrong is that it is so simple and yet the consequences of it being true are so disproportionate to its simplicity--it can't be that easy. But this sense doesn't mean the argument isn't true or that it cannot be compelling. In any case, people have tried and keep trying to figure out if it is indeed valid--and even people of faith differ.
Kant argued that existence isn't a property of a thing, so you can't really "add" it to the concept of that-than-which-no-greater-can-be-thought. That's why he thinks it doesn't work (from what I can tell). The problem I have with this objection is that I don't think he establishes that existence is not in fact something that can be added, even if it isn't a formal property. We all understand the distinction between thoughts and things that exist "in reality," as Anselm himself points out. Kant says that this existence is empirical, something involving the senses that we experience, which is of course a very subjective point of view.
I would say rather that existence is something that can be experienced, and further, even science postulates the existence of things that we as humans cannot experience directly with our senses (e.g., subatomic particles, black holes, cosmic strings, planets in other solar systems, and so on). We can only experience them indirectly and reason about their existence from the observations we make. So things can "have being" without our experiencing them (or even potentially experiencing them) directly.
We implicitly give more value to things that we perceive to have being, rather than just concepts in our heads. So whether or not existence is a proper property or not, it is something that is valuable that not all concepts have. And we think it is better to have this existence than to be thought only. So in this respect Anselm's argument still holds sway.
The other objections I've seen all hinge around disputing the premise or definition of that than which no greater can be thought. One objection, already noted, is applying that concept to things in particular categories--islands, cats, etc. Even though this seems an obvious objection, it is also the most easily refuted. The proposition of the argument is that-than-which-no-greater-can-be-thought without further qualification. It's not the greatest island--we can conceive of something greater than an island. It's not the greatest cat, or human, or computer, or any particular type of thing. It is the greatest thing, across all categories, surpassing all categories. I would suggest saying "infinite good" might help understand why this objection doesn't apply.
Good is what is desirable. Obviously, we each have our own ideas about what is desirable, which is why the infinite qualifier is important--it easily accounts for all of our particular goods that we have in mind and surpasses them. And this ties in with the discussion above about existence--we generally perceive existence as a good, and that's why we can add it to (in a sense) the goodness of the infinite good concept--wouldn't it be better for the infinite good to actually exist? Of course!
The real trouble, if any exists, lies in the assumed definition being a definition of God. Let's set aside the question of God (and all the various ideas we have attached to that concept) and instead just consider the definition Anselm provides as referring to "some thing." Science fiction has done a good job in helping us to approximate this kind of thinking--offering up aliens (like the Q in Star Trek) who are for all intents and purposes like what many conceive of as God (or at least gods). Anyways, the point is that if we can suspend judgment on whether or not the definition pertains to "God" and just consider it at face value, it seems easier to accept the argument as valid, and as such it is not, as Rev. Neal asserts, a faith-based proposition.
On this count, it seems to me that the argument aptly proves the existence of this being, logically speaking. The trouble, as I said, comes when we make the jump from from such a being to any conventional notion of God, or more specifically, the God of the Christian religion. I will not deny that God, in the proper Christian sense, is an object of faith. I would say this is indeed by definition, and dogmatic in the Christian faith. And ipso facto, there are no proofs for his existence. The can be, on the other hand, proofs that lead one to faith in God because they demonstrate truths about him and/or our relationship to him.
This is not irrational, as faith is popularly conceived to be. It is not against reason or without reason--quite the opposite in fact. Faith is an act that uses reason as a launchpad. In function, this is not different from other beliefs that people hold--they are things for which they have little, if any, direct empirical evidence for and yet accept as true, because they seem to make sense in relation to or follow from other things they have experienced, because someone they trust/an authority told them, because it is commonly held by people they know, etc. The difference is in the object--faith is belief in the Divine. Faith can can be based on any or all of these, as well as on the ontological argument.
As for Anselm's argument, it does seem to prove the existence of the being that he posits. For those who do believe in God, they would associate the posited being with God, at least as a way to understand and think about God. For those who do not believe in God, I am not sure how they might grapple with the existence of such a being. For all of us, what bearing does this being have on us? What should our relationship to it be (if any)? Should we ignore it, or should we try to understand it better? How might we understand it better?
If we can accept the existence of such a being, and it seems to me that the ontological argument makes a strong rational case for doing so, then it seems to follow that we need to come to terms with its existence somehow. (Or we can choose to simply refuse to acknowledge the validity of the argument without demonstrating what's wrong with it! ;) ) If we choose to grapple with it, there are further proofs that lead from an initial concept of God such as this to the Christian conception of God, the Trinity, and Faith. I myself have followed them, and they are rational, certainly as rational as any reason I've seen to not believe in God. And in my estimation, the reasons for faith are more compelling, taken as a whole, than those for the lack of it.