"So there remains only the idea of God: is there anything
in that which couldn’t have originated in myself? By the word
‘God’ I understand a substance that is inﬁnite, eternal, un-
changeable, independent, supremely intelligent, supremely
powerful, which created myself and anything else that may
exist. The more carefully I concentrate on these attributes,
the less possible it seems that any of them could have origi-
nated from me alone. So this whole discussion implies that
God necessarily exists."
This is a deceptively simple little argument. The basic idea it that my concept of a perfect being is so rich and expansive in its content - indeed, its representational content is infinite and perfect -- that I cannot come up with it on my own. In fact, no being less than a perfect being could create it, as the cause must be adequate to the effect. Therefore, God must exist in order to cause my idea of God. We can standardize the argument as follows:
1. If I have a concept of a perfect being, (all-knowing, all-powerful, perfectly good, absolutely independent of everything else, the creator of everything else, etc.), then a perfect being exists as its cause.
2. I have a concept of a perfect being.
3. Therefore, a perfect being exists as its cause. (from 1 and 2, Modus Ponens)
What to make of this argument? Well, it's valid; so if the premises are true, then the conclusion follows of necessity. Therefore, the only way to rationally resist the conclusion is to find an undercutting or rebutting defeater for one or more of the premises. Well, why are we supposed to accept the premises?
Premise 1 says that the presence of the concept of a perfect being entails that God exists as its cause. And as we saw above, the reason is that the concept of God is "too big", so to speak: its representational content (that which represents an unlimited, perfect being) is so great that I'm not up to the task of creating it. Indeed, the same holds true of any being less than an unlimited perfect being. For if some such being caused my concept of a perfect being, the question would arise all over again: how did that being come up with their concept of a perfect being? And since causes must be adequate to their effects, only a perfect being is sufficient to cause a concept of a perfect being.
Premise 2 says that I have a concept of a perfect being. What's the evidence that I have such a concept? Answer: introspection; that is, I introspect on my own thoughts, and find that I have such a concept.
Thus, Descartes' argument is valid, and the evidence for the premises is (perhaps surprisingly) nothing to shake a stick at. However, if you're like me, you feel uneasy about premise 1. The premise is false if, and only if, there is at least one possible case where one has the concept of a perfect being, and yet no such being exists as its cause. Thus, if one can come up with an account or "recipe" of how one might construct one's concept of God without God's help, then one thereby has a defeater for premise 1, and thus a rational basis for resisting the conclusion.
Descartes anticipated that critics would try to come up with such recipes. Indeed, he offers several such recipes, and then argues that each is inadequate. We'll consider the two most important recipes: what I'll call The Building Up Recipe and The Negating Recipe.
The Building Up Recipe can be broken down into two main steps:
Step 1: Conceptualize yourself as a limited, imperfect being.
Step 2: Successively build up your self-concept in your mind until it lacks its limits and imperfections.
According to Descartes, the Building Up Recipe is a failure. For one could never complete step 2. For that would be roughly equivalent to counting to infinity, and you can’t create an actual infinite by successive addition.
What about The Negating Recipe? Like the Building Up Recipe, this strategy can be broken down into two main steps:
Step 1: Conceptualize yourself as a limited, imperfect being.
Step 2: Negate the constituents of your self-concept:
But this concept is equivalent to that of an unlimited, perfect being. Thus, this strategy avoids the problem of successively traversing an actually infinite series of "build-ups" of your self-concept by doing it all in one stroke, i.e., merely by the simple act of negating its constituents.
According to Descartes, The Negating Recipe is a failure as well. For your concept of unlimited perfection is prior to your conceptualization of yourself as a limited, imperfect being. For you would never see yourself as limited and imperfect unless you first had a standard of unlimited perfection with which to compare yourself. In effect, then, you already require your concept of God to even get to step 1 of The Negating Recipe(!).
It turns out that it's harder to say what's wrong with Descartes' argument than one might have thought at first glance. If we are to rationally resist premise 1, we need a recipe for constructing our concept of God without God's help. But we've seen that this isn't as trivial as it seems. However, Mackie has made some remarks that suggest a way to modify The Negating Recipe so as to generate our concept of a perfect being. His fundamental point is that in order to realize that you're limited and imperfect, you may well require a higher standard with which you can compare yourself. However, it's not clear why the standard needs to be unlimited and perfect. For it seems that all one needs is a standard that's limited, but at least a bit greater than yourself. Thus, once you get such a standard, you can then run through step 1 of The Negating Recipe, and from there, follow on to step 2 by negating the constituents. And the result is the concept of an unlimited, perfect being.
For my own part, I think there's a way to modify The Building Up Recipe as well. For consider how sets are constructed in set theory. One way is by listing the members to belong to the set. Another is to offer a description of the things to belong to the set. But a third method is via recursive definition. Thus, one can construct an infinite set by taking the number 1, and then prescribing that for any natural number n, the successor of n is in the set. (One would also need a closure clause that nothing else is in the set.) In this way, an infinite can be constructed in just a few strokes, without the need to enumerate each element. Similarly, perhaps one could recursively define God's unlimited attributes.
 Descartes, Rene. "Third Meditation", in Meditations on First Philosophy, transl. William Bennett. http://www.earlymoderntexts.com/pdfbits/dm2.pdf