Gods' relation to logic is quite interesting. For St. Augustine, the truths of logic exist in the divine intellect. On this view, truths of logic are not invented or discovered. Augustine summarizes his view as,

"Ideas are the primary forms, or the permanent and immutable reasons of real things, and they are not themselves formed; so they are, as a consequence, eternal and ever the same in themselves and they are contained in the divine intelligence"

On this view, it doesn't seem God could make 2+2=5. That is because, in scholastic thought, mathematical and logical essences were independent of God's will, but were entirely dependent on His essence. Aquinas gives a succinct indictment of logical voluntarism in chapter 25 of Summa Contra Gentiles:

"Since the principles of some sciences, such as logic, geometry, and arithmetic, are drawn solely from the formal principles which constitute the essences of things, it follows that God cannot do anything which conflicts with these principles: thus, he cannot make a genus not predicable of its species, or bring it about that the radii of a circle are not equal, or that a rectilinear triangle should not have its three angles equal to two right angles"

Later thinkers, such as Descartes, would answer the affirmative. Descartes, in a letter to his friend, wrote:

"God had established these truths He could change them as a king changes his laws. To this the answer is: Yes he can, if His will can change.' But I understand them to be eternal and unchangeable —I make the same judgment about God. But His will is free.—Yes, but His power is incomprehensible.'"

There are two ways in which such a view differs from the more traditional, Scholastic view. The first is its Platonic aspect : the mathematical essences are distinct from the essence of God. This would have been rejected by the medieval thinkers. Second, is its voluntarist aspect: the mathematical essences are under the control of God's will. In Descartes view, it was God's benevolence that excluded God from changing his legislations. God would be a deceiver, Descartes thought, if — while giving me such a nature that I perceive these laws as immutable — he had also decreed that the laws were to change. It follows from God's nature that these truths are immutable, even though God was wholly free when establishing them.

I don't think Descartes' view is coherent. If God established the truths of logic, there would be points of the global causal order in which the truths of logic didn't obtain and, consequent to God's willing, did. This seems quite incoherent. Surely if God exists, he is identical to himself at all points of the global causal order. But this seems to suppose the law of identity. Hence, it seems quite senseless to say such a law of logic was not instantiated at all points of the causal order.

I don't have much confidence in the Scholastic view, so the real choice lies between Platonism and various forms of constructivism. Contemporary Platonists insist that insists that truths of logic are not truths about the contents of any mind, divine (as the Augustinian would have us believe) or mundane (as the conventionalist would have us believe). Michael Dummett gives a nice summary of conventionalism. As he writes,

"According to conventionalism, all necessity is imposed by us not on reality, but upon our language; a statement is necessary by virtue of our having chosen not to count anything as falsifying it. Our recognition of logical necessity thus becomes a particular case of our knowledge of our own intentions."

For conventionalism, as Dummett explains, our knowledge of logic and mathematics reduces to knowledge of mathematicians and logicians' intentions. If the conventionalists are right, God's knowledge of the a priori disciplines reduces to his knowledge of his creatures: in particular to his knowledge of the powers, activities, and decisions of human beings. The later Wittgenstein describes this view quite vividly in Remarks on the Foundations of Mathematics:

"Suppose that people go on and on calculating the expansion of it; so God, who knows everything, knows whether they will have reached '777' by the end of the world. But can his omniscience decide whether they would have reached it after the end of the world? It cannot. I want to say: Even God can determine something mathematical only by mathematics"

I personally, like Descartes, want to reject the Scholastic view and hold that logical truths distinct from the essence of God. But I also want to reject voluntarism or conventionalism; the necessity of mathematics or logic has nothing to do with God's creative action nor with the intentions of interested logicians and mathematicians. Hence, I take the contemporary Platonist perspective that the truths of logic are not truths about God's intellect or essence; truths of logic are simply truths about eternal, extra-mental, abstract entities. This sort of classical Platonism does seem to be a bit hard to reconcile with theism; it posits infinite realms of being which are independent of God, exist just as necessarily and uncreatedly as him, and were established independent of him. For me, God's relationship to Platonic truths may be mysterious, however, is no less mysterious how we humans can come to know them by means of the artful manipulation of symbols. I personally find Peter van Inwagen's work on how theism and uncreated, necessary abstract objects may be reconciled helpful. His essay,"God and other uncreated things", while it doesn't resolve all the tension, certainly leaves one with a sense of optimism.