The essence of the doctrine of the trinity, from memory, is that God is one “nature” subsisting in three “persons”. God is the father, the son and the holy spirit; yet there is one god, not three gods. The father is indivisible, the son indivisible, the holy spirit indivisible; yet there is one indivisible, not three indivisibles. Likewise, the father is incomprehensible, the son incomrehensible, the spirit incomprehensible; yet there is one incomprehensible, not three incomprehensibles. The Father is not created, the son is not created but is begotten of the father, and the spirit is not created but proceeding, yet there is one eternal. And so on.
Now, it is not really possible to understand any of this stuff. This is not controversial – the nature of God is admittedly a mystery. In fact, the doctrine of the trinity as it has been stated over the years is simply a list of things that one is *not* to believe. One is not to suppose that the son is a being created of the father – that is Arianism and a heresy. Neither should one imagine that there is one god who presents different faces to the world like an actor or a pupetter – that is modalism. And so on.
What we are left with is “one nature, subsisting in three persons”, which is a declaration startlingly void of content. What does “subsisting” mean, anyway? I know what a “person” is, more or less, but what’s a “nature”? How is “proceeding” from the father different from “begotten” of the father? Well, once again, we just don’t know.
This raises a couple of fun questions. What exactly does it mean to “believe in the trinity”, if the doctrine itself is basically void of any content aside from the fuzziest “well, there’s three of them but there’s really only one”? How can a person believe something when there’s effectively nothing to believe? Does “belief in the trinity” mean, has it ever meant anything more than that you are prepared to affirm that you believe it?
And how did the ealy christians know that the one nature “subsists” in three persons? Perhaps ‘subsists’ is simply the word that we use to describe the relationship, and that’s what it means. The word is not meant to convey meaning but is simply a label for the incomprehensible. We could equally well have used ‘spongs’.
As far as I can tell, when a person claims to belive in the doctrine of the trinity, they are saying nothing in particular.
PS: I’m not entirely certain what “Spirit” is either. As far as I can tell from the bible, it is the animating force of physical life. “Breath”, both in hebrew and in greek. The bible and the NT in particular teaches that man’s nature is not triune, but dual.