Here's something I just wrote in a different context, but realized might be useful here:
God works with his people in a variety of ways. First Century Judaism had many ways of talking about his presence. These included the Holy Spirit, and a set of what are sometimes called intermediate forms. Torah, Word, and Wisdom were spoken of as if they were separate entities, though it may be that there were more like extended metaphors. John 1 reflects this, but you'll see many places in the NT, some that are closer to 1st Cent Wisdom theology than John's Logos. (A personified Wisdom developed from the descriptions of Wisdom in Proverbs.) But all of this expressed the presence of a God who without them might tend to look isolated from us.
The NT refers to Father, Son and Holy Spirit a few places, and in more places refers to the Holy Spirit and the Son individually.
Now these could have been, and at times were, thought of simply as ways that we experience God, and thus not any distinction within God. I think there are both good and questionable reasons why it didn’t end there. Historically, I think it was for a bad reason: There was a strong commitment to God being impassible, unable to suffer. The ideas that later led to the Trinity and Incarnation seem to have taken over as a result of a controversy in which the earlier ideas led to the Father suffering. Even though it was only through his human form, that was considered unacceptable. Treating the Son as having a certain distinction preserved that Father from suffering.
From that point there were a set of controversies about how to think and speak about this distinction. Since everyone agreed that there’s only one God, but there has to be some kind of distinction between Father and Son, the term hypostasis was adopted to refer to the distinct aspects, and the term ousia to refer to what is common. (The Holy Spirit wasn’t critical in formulating the Trinity. Everyone understood that it was there, so it came along as the third person. Note that the original version of the Nicene Creed has a paragraph on the Father, a paragraph on the Son, and then “and in the Holy Spirit.”)
My understanding is that this really redefined two existing Greek words. You can’t learn much about the Trinity by looking at how those terms were used in Greek philosophy, since they were close to synonyms. But the Fathers understood that existing philosophy wasn’t up to describing God. So while they reused existing terms, they made them technical terms for what is distinct and what is common about God. I’m not sure they were always so clear that that’s what they were doing, but in retrospect it seems true. I believe lots of unnecessary debate could be avoided by saying simply that hypostasis is how we refer to the distinction between Father, Son and Holy Spirit, and ousia the nature of God himself.
I would argue that there’s another way to look at it that isn’t quite as strongly based on impassibility (which I think is clearly unbiblical). If Jesus shows us God, what kind of God does he show us? It’s not the pure unmoved mover. The Muslim God, and certain Jewish concepts of God, couldn’t possibly appear in a form that suffers. I would suggest that if Jesus shows us God, this implies a certain complexity about God, that’s he not just the masterful creator, but the obedient son, and thus experiences the relationship of love. Thus the idea that the three persons are just ways in which we experience God isn’t good enough. It characterizes God himself. (Note that this is close to Augustine's explanation. He says that the only distinction between Father, Son and Holy Spirit is relational. i.e. that there's only enough distinction to allow the relationship of love to exist within God. We don't want to say that God was unable to love until he created humans to love. The article on the Trinity in the Catholic Encyclopedia says that God is one, but experiences in a three-fold way.*)
But I’m far from clear that speaking of three hypostases is a natural way to refer to this. One problem I see with the traditional language is that it makes Father, Son and Holy Spirit look too much the same, like three of the same thing, which ends up being hard to distinguish from there Gods. But the Biblical basis isn’t so symmetrical. Son and Holy Spirit in the Bible are ways that God is present with his people. I think it is a mistake to make the Trinity about the number three. Rather, I would use more complex and less philosophical language. I would say that Biblically the Son and the Holy Spirit as ways in which God is present with us, but these are not just how we experience God, but that they truly reflect aspects of God himself. I would say that I'm Trinitarian, because I do recognize that Father, Son and Holy Spirit represent something about God himself and not just our experience of him, and I would certainly never support the idea that the Son is a separate and inferior entity (i.e. Arianism). But I think the Athanasian Creed goes too far beyond the Biblical ways of thinking and speaking (not to mention the anathemas, which have their own problem). The Nicene Creed doesn't have this problem.
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* Here's the actual phrase: "the same mind will have a three-fold consciousness, knowing itself in three ways in accordance with its three modes of existence."
