The question then, is whether this formula is compatible with the religion of Israel. Trinitarian monotheism states that there is one God, and yet three persons in that one God. One of the more unusual philosophical argument I have heard was from a Hasidic anti-missionary, who argued that Leibniz' Law prevents such an understanding.
Leibniz' Law
If two objects have all the same properties, then they are the same object. This is closely related to the indescernibility of identicals, which is stated as follows:
Indescernibility of Identicals
If two objects are identical, then they have all the same properties.
So now the problem arises. If God the Son has the property of "being incarnated" and the Father does not, does this make them different beings, and hence, prevent us from affirming monotheism?
Aquinas used divine simplicity as his way out. As one Thomist stated:
Relations which result from the mental operation alone in the objects understood are logical relations only (inasmuch as reason observes them as existing between two objects perceived by the mind).So, Thomists think of the Father, Son, and Spirit as real relations within one being. Divine simplicity is difficult to understand, and I recommend the Wikipedia article to see the difficulties with it. The short answer is that if divine simplicity is true, then God does not have any properties, and even our language cannot apply to God. So did that previous statement apply to God? If it did, then it's false, for we have found a counterexample. If it did not, then it does not apply to God, and so is false as well.
Those relations, however, which follow the operation of the intellect, and which exist between the word intellectually proceeding and the source whence it proceeds, are not logical relations only, but are real relations (inasmuch as the intellect and the reason are real things, and are really related to that which proceeds from them intelligibly: as a corporeal thing is related to that which proceeds from it corporeally). Thus paternity and filiation are real relations in God.
With that as an unlikely choice, we can think of another solution. What is God? Well, we generally think of God as an unembodied mind. What does a mind have? It seems that a mind has mental faculties such as reason, volition, will, memory, consciousness, and other properties, and has these essentially. We also think of God as having omnipotence, omniscience, and omnibenevolence. The problem is that none of these things have the same properties. We cannot say the same thing of God's will as we can say of God's memory or power, or consciousness.
Do these plural attributes mean that there are many Gods? Not unless we think of faculties as distinctly separate objects. So as long as we do not consider God's center of self-consciousness as a different object from his memory, we still have monotheism.
Now consider the Trinity doctrine, that God is one mind or soul with such strong mental faculties that he has three centers of self-consciousness. It doesn't seem to commit us to Tritheism any more than God's separate will, memory, and consciousness do.
One tactic by anti-missionaries is to bite the bullet and say that God did not have any properties before creation. They would say that God's attributes are descriptions of him after creation. The problem with this view is that God is not dependent upon creation. He could have chosen to refrain from creation. If that were the case, would God be loving, merciful, kind, powerful, or would even exist? If these properties are time-dependent, and time is dependent upon creation, then the answer is no. This means that God is not essentially loving, merciful, kind, or even a necessary being!
As Psalm 90:2 states: "From everlasting to everlasting, you are God." The rabbis affirm this verse in the Psalms as from God. If they believe that before the world existed, the term "God exists" is undefined, it is inconsistent of them to demand that "The Son is begotten of the Father" or "The Spirit proceeds from the Father" are defined.
This is just to admit that the uni-personal God of Judaism and Islam cannot be the perfect being of perfect being theology. Such a God cannot be that than which nothing greater can be conceived. For we can conceive of a being that is greater: one that is essentially loving, kind, and existing.
The problem only arises that to talk about a separate person is to talk about a separate being. We make the distinction between "being" and "person" all the time. This desk is a being, but certainly not a person. Since a center of self-consciousness is a property of the mind and not a separate being, we can state that one mind with 100 centers of self-consciousness would be one being and 100 persons. It might be hard to grasp intuitively, since we are not multi-personal beings, but it's hardly incoherent.