For, it is one thing for an object to be in the understanding, and another to understand that the object exists. ...
Hence, even the fools is convinced that something exists in the understanding, at least, than which nothing greater can be conceived. For, when he hears of this, he understands it. And whatever is understood, exists in the understanding. And assuredly that, than which nothing greater can be conceived, cannot exist in the understanding alone. For, suppose it exists in the understanding alone: then it can be conceived to exist in reality; which is greater.
Therefore, if that, than which nothing greater can be conceived, exists in the understanding alone, the very being, than which nothing greater can be conceived, is one, than which a greater can be conceived. But obviously this is impossible. Hence, there is no doubt that there exists a being, than which nothing greater can be conceived, and it exists both in the understanding and in reality.
- Anselm, St. Anselm: Basic Writings. Translated by S.N. Deane (Second edition; Open Court Publishing, 1962). As quoted in Diogenes Allen and Eric O. Springsted (eds), Primary Readings in Philosophy for Understanding Theology (Louisville, Kentucky: WJKP, 1992)
Anselm's ontological argument is one of the classical proofs for the existence of God. It can be summarized as saying that God is the thing of which nothing great can be conceived. Since everyone can conceive of God as being the being of which there is nothing greater than, therefore God exists.
There are however a few problems with Anselm's ontological argument that I can see. One problem with Anselm's ontological argument lies in its assumption of reality being ontologically superior than mere thought. Upon what basis can a real greatest conceivable being be considered as superior over a virtual greatest conceivable being? For it is not agreed among philosophers or society in general that reality is superior, and in what way superior such that reality is preferred over that which is solely conceptual.
The second great hole in Anselm's argument is that it assumes that everyone is agreed as to the existence of qualities comparing the greatness of things conceived. However, why must such "grading on a scale" exist? How do we compare apples and oranges? Is an apple more of a fruit than an orange?
Here the problem lies not so much in subjectivity or relativism but rather in Anselm's assumption that there is a checklist of qualities which denote what is great. However, among two "conceivable" objects there may have different qualities which are just different, and good in their way. Just as a car is not better than a toothbrush since there have different functions and purposes and thus are evaluated differently, so two conceivable objects may have different qualities which are not comparable to each other.
Multiplying this across all conceivable objects, coming up with the greatest conceivable being is just not possible. And if it is not possible, then how can the Ontological argument work?
In his case, what Anselm had done was "cheating" — in taking Christian or Platonic qualities and assume them for the ontological argument, yet the choice of such qualities are not questioned at all.
In conclusion, based upon these two fallacies, Anselm's ontological argument is not valid. It presupposes either a Platonic or Christian-type worldview and will fall apart from it.