Believe then understand

Nice article, Peter! One nuance I'd like to add: you say of the Christian mysteries that "We can at least establish that they involve no impossibilities." The view of Aquinas, as explained by Reginald Garrigou-Lagrange, is actually subtler: we cannot prove the Trinity to be non-contradictory but we can show that it cannot be proven to be contradictory.

For Catholics this is especially important to get right with regards to the Trinity because as the ground of all being, God contains nothing that is unnecessary. That means that anything that can be rigorously demonstrated to be possible about God, must be not only possible but necessarily true. So if a Catholic says we can prove the Trinity is possible he is implying that God can be known to be a Trinity by reason alone. But if we can know even the supernatural mysteries of God by reason then the distinction between supernatural and natural, between God and man, breaks down. We can know God's attributes by reason because they are attributes we can naturally participate in, but what belongs to the intimate life of the Deity qua Deity (not qua good, powerful etc) is unknowable by the human intellect.

The same principle (it cannot be proven possible but we can see that it can't be proven impossible) applies to things like the Incarnation too, but since the Incarnation is contingent, a free act of God, saying it can be proven possible at least doesn't cause the same massive problem as with the Trinity.