Okay, I'm a day late, but I thought it would be amusing to consider the justification of the Trinity. To many folks the whole idea seems wildly contradictory, and makes one wonder why so many very smart people had no problem with it. There are two possibilities. Those very smart people weren't so smart or we just don't get it.
The latter is not beyond possibility. The folks of those days operated with different categories of thought than we do. Many of them used the system known as Neoplatonism, but for the most part they used the system known as Aristotelianism. To moderns, the differences are hard to see, primarily because they are both different from Scientism. To post moderns, all three are Whatever.
It is interesting that the founder of Neoplatonism, Plotinus, deduced a Trinity from Platonic principles, which he called the One, the Intellect, and the Soul. If you note a similarity between that and the Christian Trinity of Creator, Word, and Spirit, you will not be the first. Plotinus worked while Christianity was gaining popularity, but before the terrible persecutions. (Plotinus also gathered twelve disciples around him, one of whom [Porphyry] gathered his teachings into books and wrote a biography of his teacher.) Neoplatonism was very popular among early Christians. Thus, when we read that Hypatia of Alexandria "succeeded to the school of Plato and Plotinus," we should understand 1) she was teaching a philosophy that most Christians found agreeable and 2) she would not likely have been an atheist.
Plotinus' arguments can be found here, here, and here.
The Aristotelian argument, as articulated by Aquinas, runs differently. (We must assume certain things as already having been demonstrated in prior proofs.)
Absent creation, He IS knows Himself. As the Knower, He is called the Father. Knowing is an act of conception. When we know a thing, we conceive it. If He IS knows Himself, he must conceive himself. So, as the Known, He is what we call the Son. But since the conceptions of the intellect are conceived in words, the Son is also called the Word. (Also: the only-conceived.)
Volition is the desire for the fruits of the intellect. The Desirer is the Knower, the Father, since one cannot desire what is not known. Desire proceeds from the desirer to the desired and returns to its source. As the Desired, He is called the Spirit, which is said to proceed from the Father. But since the act of desiring is called love, the Spirit is also called Love.
Since Existence Itself is purely actual, each of these -- Father, Son, Spirit -- is fully actual, and must therefore exist as persons in their own right. But since He IS is One, these three Persons are one Being.
In graphical form:
Act of Intellect............................Act of Volition
Son←conceived--Father--proceeds→Spirit
None of this will convince a Jehovah Witness or a Unitarian, but the Middle Ages was bug-nuts for logic and reason.
Figure 1. The rose window shown is that of old St. Joseph's, now Our Lady of Mercy parish, in Easton PA. The medallion at the center of the rose is unusual in that it portrays the Father and the Spirit as participating in the crucifixion of the Son. They are portrayed with arms or wings outstretched. The triptych portrays Pentecost.
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