Indynoticer when he grows up:
Here is a primitive argument for the Trinity: The world must have begun, as evinced by the fact that every finite sequence of events requires an actual beginning, and every ordered series of efficient causes requires a first efficient cause. I know St. Aquinas disputes whether reason alone can demonstrate the temporal beginning of the world, but I think there is sufficient evidence for it. Even if this is not the case temporally, it is true ontologically: the world is continuously sustained by some creative act, and this, at least, can be demonstrated by reason. Without such a necessary foundation, the contingent world would not exist.
Without relying upon revelation, then, we can maintain that God created, or creates, the world from nothing—as only that which exists outside of time could have created (and time is the measure of motion and change).
If God created from nothing, then He could not have acquired His actuality from the act of creating. God does not become more actual by creating, since the act of creating presupposes the created. God is immutable, and immutability is one of His perfections; therefore, He cannot become more or less actual through the existence or non-existence of creation. This follows from God’s immutability, which is an aforementioned condition of His capacity to create a mutable world.
If the God who does not create the world is the same God as the one who does (modally), then creation must be a selfless and gratuitous act. If creation is selfless and gratuitous, then selflessness and gratuity cannot be accidental to God, but must belong to His essence.
It is impossible, however, for a merely solitary, monadic being to be essentially self-giving and gratuitous. A being can only give itself if there is another to whom it gives itself. But because only God exists necessarily, this other cannot be another being alongside God. God must therefore give Himself to Himself as Other. He must differ from Himself relationally while remaining essentially identical to Himself.
The Son of God is this Other of God: He is God, but He is selflessly given by the Father. If the Son is selflessly given Himself, then He must give Himself back, for He does not receive Himself merely as a “gift,” but as the act of “giving.” For instance, one might receive a gift from another and store it away; but if one receives the act of giving from another, one must oneself give what one possesses, since the act rather than the object has been received. The Son, being absolutely simple, has only Himself to give, and therefore returns Himself to the Father.
The selfless act giving shared by the Father and the Son is not something external to God, nor something added to Him. This mutual love of the Father and the Son is what we term the Holy Spirit.
Therefore, if the world was created, as can be demonstrated at least in the ontological sense, the Trinity is not arbitrary—but a sufficient (or at least fitting) condition of God’s aseity and immutability.