The Trinity is actually implied in the Divine Name: “I Am that Am” (I would argue that the filioque is as well, but no need to open that can of worms).
What we see with the Divine Name is three key ingredients, all of which underly the Trinity. The “ingredients” are eternity, existence, and personhood. Of these three, eternity and existence are relatively simple and don’t require much comment. The third, however, is where everything turns. It’s also important for this philosophical exercise to view the Divine Name as the most perfect human “expression” of the Divine Essence (recognizing, of course, that there is really no such thing as sufficiently dealing with such a huge matter; I mean this more as a summary of God’s revelation to Moses about His identity).
I would argue that personhood is fundamentally built on three things: being, knowledge, and will. Philosophers traditionally expressed this in such ways as “individual, knowledgable subsistence”, but those three aspects are the gist of it.
First we have being: the Divine Essence Is. Since the Divine Essence is Personal, and person includes the notion of being, the Divine Essence is eternally a Person by virtue of It being. Now, the Divine Essence being is what we call the Father, the unoriginate source of all Deity.
In being, and in being eternal, the Father knows from all eternity, as knowledge is the second fundamental aspect of Personhood, which is a fundamental aspect of the Divine Essence. Now the notion of “knowing” contains two opposite terms: knower and known. In this case, for the purpose of identifying the terms, the knower is the Father, and the Known is the Divine Essence. In us there is nothing spectacular in saying that a person knows themself, since our Essence is neither necessarily existing, nor is it necessarily personal (it can be abstracted fully before we even exist, eliminating the “being” part of person) but with the Divine Essence things are a bit more complex because of the “definition” given in the Divine Name: if the Divine Essence is Personal, as the Name implies, then the Essence Known is Personal; and if the notion of knowing implies an opposite term from knower (the Father), then we have an opposite Person. This is by necessity because of the inherently Personal element of what is known, and the opposing terms of knower and known. We now have the Person of the Son, who incidently is also called the Word which is the ancient philosophical term for a mental conception. This Person is still “I Am that Am” because He is the Divine Essence, and there are not multiple “Essence Knowns” because there is only one knowledge in the notion of Personhood, and the Divine Essence is Personal. We have a multiplication of Essence, and a distinct Person, without a division of Essence. The Son contains everything in Him that the Father does because He shares the Essence, with the exception of being the Father since “known” is opposite of “knower”.
Likewise with the Holy Spirit, who is the notion of the “will”. There are two terms: willed and willer. And what is willed from all eternity but the Divine Essence? We say this not because the Divine Essence is a creature, but because existing persons necessarily “will themselves”, as I will to be Ghosty even if I don’t produce myself. Again, the Essence is Personal, and It is willed from all eternity, and will is opposite from willer as distinct terms. And again, there is only one “Willed” Person of the Divine Essence because there is only one will contained in the notion of Person. So what is Willed but the Divine Essence, complete in every way except that It isn’t the Willer (the Father). We have once again multiplied the Divine Essence (I Am that Am) without dividing it, albeit we have a distinct Person yet again by virtue of opposite terms.
continued…