A less popular argument for atheism, than the problem of evil, is the problem of divine hiddeness. God's existence, it is said (even by believers), is not as clear as it could be. But yet God wants us to believe in Him. This problem may be expressed more formally, as a reductio ad absurdum, as follows:
0. God exists.
1. God wants all men to believe He exists.
2. Since God wants this, He would have made Himself clearly known.
3. There are educated-rational men, who do not believe in Him.
4. Therefor, there is no God.
So then, a reduction to absurdity, such as the above, starts out assuming the truth of what is to be disproved, the zeroth premise, then shows a contradiction is derivable from the original hypothesis. The conclusion, (4), contradicts the original assumption, (0), and so, if the proof is sound, we must reject the original assumption. In other words, if the above proof succeeds, God does not exist.
But, I do not think the above proof is sound. Let's take a closer look to find out why. First, in premise 1, the term "believe" is ambiguous. According to the letter of James, in the Bible, Satan himself believes in God. But we would hardly expect that the actions of the devil are pleasing to God. There is certainly a difference between mere mental ascent, such as devils possess, and saving faith such as Christians possess. It seems that saving faith is what is in view in premise one.
The reformers distinguished three distinct components of saving faith. They even had fancy Latin verbiage to label said components. I don't recall off hand what those terms are but no matter. What is sufficient for our purposes here is that saving belief is more than mere mental ascent. The something more includes an accepting trust. It is not enough to say, "I know God exists, so I guess I'll just have to make the best of it." We have to like the fact that God exists, and willingly follow Him.
But here is where the real problem comes in. Does God want everyone to choose to follow Him? I think He does. But what if we choose, of our own free will, to not 'trust and obey'? God cannot make us freely choose to follow Him. We saw this when looking at the problem of evil. So then, if God first wants us to have free will (and He does) He cannot then make us choose to follow Him of our own free will, even against our own free will, for that is self-contradictory and, hence, meaningless.
But we can go further. Premise two is demonstrably false. God could (and "could" is all we need to defuse a deductive premise) have had over-riding reasons for not making His existence more clear. God's thoughts are as high above ours as the heavens are above the earth. So maybe (and "maybe" is all we need to defuse a deductive premise) He knows something, in His omniscience, which we, in our limited knowledge, do not know.
What is more, premise two rests on a false assumption. Namely, the assumption God has not made His existence abundantly clear. At least, this idea seems to be lurking unstated somewheres within the above proof. But this assumption is patently false. The cosmological, teleological, ontological, and axiological proofs for the existence of God, individually and collectively, make it literally undeniable that there is a God. So God has made His existence unmistakable. If the people in premise three suppress this knowledge, that is their problem, not God's.
Could God have made His presence more clear? I suppose He could have, in some sense. He could have written "God exists" on every cell, for example. However, natural theology has given us apodictic certainty that God exists. Nothing can be more plain than what been rigorously demonstrated to be absolutely certainly true.
The argument against the existence of God, then, from divine hiddeness, is manifestly unsound as we have just seen. Still, it is a helpful argument. What I mean is that when we are exposed to the argument, we are immediately reminded of all the sound arguments for the existence of God. In this respect only, the atheistic argument succeeds.