I must admit that I am more than a little puzzled by the appeal of this argument, which has taken various forms over the centuries since it was first advanced by St. Anselm in the 11th century. Some of the greatest minds have taken a crack at it, some to try and improve upon it, others to refute it.
According to the Stanford Dictionary of Philosophy, the argument goes like this:
St. Anselm claims to derive the existence of God from the concept of a being than which no greater can be conceived. St. Anselm reasoned that, if such a being fails to exist, then a greater being—namely, a being than which no greater can be conceived, and which exists—can be conceived. But this would be absurd: nothing can be greater than a being than which no greater can be conceived. So a being than which no greater can be conceived—i.e., God—exists. [italics in original]
As one of the philosophers in the radio program said, this is the most austere of the three major philosophical proofs for god, which can be classified as (1) the Design Argument, (2) the Cosmological Argument, and (3) the Ontological Argument.
The Design Argument is rich in facts. It asks us to look at all the features of the world and especially of nature and observe how well they are designed, and infers that there must be a designer, ergo god. The Cosmological Argument eschews all that and looks at just one brute fact, that our universe exists at all, and poses the fundamental question of why is there something rather than nothing (a question I discussed recently) and from that infers that god must exist in order to not have nothing. The Ontological Argument eschews empirical facts altogether and seeks to prove the existence of god using pure logic alone.
I discussed the Ontological Argument last year and expressed my skepticism about it.
I must admit, I just don’t get it. As I have said many times, I simply do not see how you can answer an empirical question of the existence of anything using pure reasoning without any supporting data. Just because you can conceive of something or because something is possible to exist cannot lead to any firm empirical conclusions as to its existence.
If this is the best argument that theologians can come up with, then god is done for.
After listening to this interesting discussion (which I can recommend because it is very thoughtful), my earlier conclusion remains unshaken.