Does Descartes succeed in proving that God exists?

No, he does not!

Descartes offers two different "proofs"for the existence of God.   He presents his primary "causal"proof in Meditation III.   He later presents a pair of "ontological"proofs in Meditation V.   Unfortunately for Descartes, the primary proof in Meditation III is fatally flawed by a reliance on a premise that is in fact false.   One of the two ontological proofs depends on the consequences of that failed proof in Meditation III.   And the other ontological proof employs a premise that is likewise simply false.

In Meditation III, Descartes attempts to demonstrate that God exists by relying on premises that he argues are "beyond doubt".   Specifically, he argues that these premises are immune to the deceptions of his Evil Demon.   The first of these is his famous "Cogito".   Descartes argues that it is indubitable that "I think, I exist".   He reasons that even under the challenge by an all powerful Evil Demon intent on deceiving him about all manner of things, the deceiver cannot possibly deceive him about the fact that he thinks and exists.   Because to be deceived is to think and to exist.   The negation of this premise is self-contradictory.  

All of the premises that Descartes employs in his Meditation III proof for the existence of God are supposedly of this nature.   The negation of these premises are, according to Descartes, self-contradictory (or would, in his worlds, involve a "manifest contradiction").   And it is beyond the power of his Evil Demon to make Descartes believe such a manifest contradiction.

Where Descartes goes wrong, however, is in his setup of his notions of "objective reality"and "formal reality".   These are concepts that Descartes defines as part of his proof in Meditation III.   A thing (like an idea) has an objective reality in virtue of the thing that it represents.   And a thing (again, like an idea) has a formal reality in virtue of its real existence.   Ideas have a formal reality only in virtue of their existence as ideas.   Whereas they have objective reality in virtue of the formal reality of the thing that they represent.   Descartes assumption is that an idea's objective reality must be caused by the formal reality of the thing that the idea represents.   His key erroneous premise is that the formal reality of the thing that is the cause of the objective reality of some idea must be greater or equal to that objective reality.  

Descartes reasons that his idea of a supremely perfect God has a supremely perfect degree of objective reality.   Hence, he further reasons, this idea of supreme perfection must be caused by a formal reality of at least as high a degree of perfection.   Hence God, as that supremely high degree of formal reality, must exist.

Where Descartes goes wrong, here, was demonstrated 150 years later by Darwin.   Darwin's Theory of Evolution by Natural Selection showed that it is not necessary that an idea's degree of objective reality must necessarily be caused by some thing that has as much formal reality.   (Although of course, Darwin did not reason in this fashion.)   In other words, the Theory of Evolution by Natural Selection demonstrated that Descartes key premise was in fact false.   Which means that Descartes primary proof for the existence of God, in Meditation III, fails the "soundness"test -- not all of his premises are true.

Having concluded that a supremely perfect God exists, Descartes draws the further conclusion that such a God would not permit Descartes to be systematically deceived.   Therefore, Descartes reasons, he can rely on the fact that all of his "clear and distinct perceptions"are necessarily true.   This further conclusion becomes the key to his first ontological proof in Meditation V.

In Meditation V, Descartes reasons that:-

(1)    whatever I clearly and distinctly perceive to be contained within the idea of some thing is necessarily true of that thing.   (In virtue of the fact that God would not permit him to be deceived about what he clearly and distinctly perceives).

(2)    I clearly and distinctly perceive that necessary existence is contained within my idea of God.

(3)    Therefore, necessary existence is truly contained within God.

(4)    Therefore God necessarily exists.

But unfortunately for this reasoning, it all depends on the key premise that God exists and would not deceive him about the truth of his clear and distinct perceptions.   And that, we saw above, is a conclusion that Descartes cannot properly draw since his proof in Meditation III was flawed.   Besides which, is the invalid slide from "necessary existence is contained within my idea of God"to "necessary existence is contained within God".   Descartes'"clear and distinct perception"is of his idea of God, not of the reality of God.   So even granting that his clear and distinct perceptions are true, that would only mean that it is true that necessary existence is contained within the idea of God.   It does not entail that necessary existence is contained within the actuality of God.   So this argument fails on two counts.

Later in Meditation V, Descartes provides the second of his ontological proofs.   He reasons that:-

(1)    I have an idea of a supremely perfect being, i.e. a being possessing all perfections, a being more perfect than which is not possible.

(2)    Necessary existence is a perfection.

(3)    Therefore a supremely perfect being (aka "God") necessarily exists.

This particular form of the ontological argument fails, of course, in the second premise.   It is not reasonable to suppose that necessary existence is a "perfection".   I can provide just as "reasonable"an argument to the effect that non-existence is a greater perfection.   Consider that:-

(1)    I have an idea of a supremely perfect being, i.e. a being possessing all perfections, a being more perfect than which is not possible.

(2)    A being that can accomplish all those things attributed to God, and yet remain non-existent is more perfect than a being that must exist in order to accomplish those things.

(3)    Therefore, non-existence must be a greater perfection than existence when it comes to omnipotent omniperfect beings.

(4)    Therefore an omnipotent, omniperfect God does not exist.

In other words, it can be demonstrated that all three of Descartes "proofs"for the existence of God are fundamentally flawed in that they rely on false premises.

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