I don't understand it either. Not believing in unicorns doesn't confirm their existence.
They were probably referring to ontological proofs
, but I don't find them very compelling.
From the link you posted:
Descartes wrote in the Fifth Meditation:
But if the mere fact that I can produce from my thought the idea of something entails that everything which I clearly and distinctly perceive to belong to that thing really does belong to it, is not this a possible basis for another argument to prove the existence of God? Certainly, the idea of God, or a supremely perfect being, is one that I find within me just as surely as the idea of any shape or number. And my understanding that it belongs to his nature that he always exists is no less clear and distinct than is the case when I prove of any shape or number that some property belongs to its nature (AT 7:65; CSM 2:45
Whatever method of proof I use, I am always brought back to the fact that it is only what I clearly and distinctly perceive that completely convinces me. Some of the things I clearly and distinctly perceive are obvious to everyone, while others are discovered only by those who look more closely and investigate more carefully; but once they have been discovered, the latter are judged to be just as certain as the former. In the case of a right-angled triangle, for example, the fact that the square on the hypotenuse is equal to the square on the other two sides is not so readily apparent as the fact that the hypotenuse subtends the largest angle; but once one has seen it, one believes it just as strongly. But as regards God, if I were not overwhelmed by philosophical prejudices, and if the images of things perceived by the senses did not besiege my thought on every side, I would certainly acknowledge him sooner and more easily than anything else. For what is more manifest than the fact that the supreme being exists, or that God, to whose essence alone existence belongs, exists? (AT 7:68–69; CSM 2:47)
So basically he is using his idea of god as proof that god exists. Because if god did not exist he would not be able to come up with the concept of god.
For some reason this reminds me of an incident from when I was a kid. My dad came home with a mounted jackalope. (http://en.wikipedia.org/wiki/Jackalope
) Anyway based on the fact that there was the head of one mounted on the wall in my house, I was absolutely convinced that jackalopes existed, after all I'd seen one. I was quite disappointed to learn a couple of years later that jackalopes do not exist. This example shows that the concept for something can exist without there being a true physical (or metaphysical) being.