Objection 1. It would seem there is not trinity in God. For every name in God signifies substance or relation. But this name "Trinity" does not signify the substance; otherwise it would be predicated of each one of the persons: nor does it signify relation; for it does not express a name that refers to another. Therefore the word "Trinity" is not to be applied to God.
Objection 2. Further, this word "trinity" is a collective term, since it signifies multitude. But such a word does not apply to God; as the unity of a collective name is the least of unities, whereas in God there exists the greatest possible unity. Therefore this word "trinity" does not apply to God.
Objection 3. Further, every trine is threefold. But in God there is not triplicity; since triplicity is a kind of inequality. Therefore neither is there trinity in God.
Objection 4. Further, all that exists in God exists in the unity of the divine essence; because God is His own essence. Therefore, if Trinity exists in God, it exists in the unity of the divine essence; and thus in God there would be three essential unities; which is heresy.
Objection 5. Further, in all that is said of God, the concrete is predicated of the abstract; for Deity is God and paternity is the Father. But the Trinity cannot be called trine; otherwise there would be nine realities in God; which, of course, is erroneous. Therefore the word trinity is not to be applied to God.
On the contrary, Athanasius says: "Unity in Trinity; and Trinity in Unity is to be revered."
I answer that, The name "Trinity" in God signifies the determinate number of persons. And so the plurality of persons in God requires that we should use the word trinity; because what is indeterminately signified by plurality, is signified by trinity in a determinate manner.
Reply to Objection 1. In its etymological sense, this word "Trinity" seems to signify the one essence of the three persons, according as trinity may mean trine-unity. But in the strict meaning of the term it rather signifies the number of persons of one essence; and on this account we cannot say that the Father is the Trinity, as He is not three persons. Yet it does not mean the relations themselves of the Persons, but rather the number of persons related to each other; and hence it is that the word in itself does not express regard to another.
Reply to Objection 2. Two things are implied in a collective term, plurality of the "supposita," and a unity of some kind of order. For "people" is a multitude of men comprehended under a certain order. In the first sense, this word "trinity" is like other collective words; but in the second sense it differs from them, because in the divine Trinity not only is there unity of order, but also with this there is unity of essence.
Reply to Objection 3. "Trinity" is taken in an absolute sense; for it signifies the threefold number of persons. "Triplicity" signifies a proportion of inequality; for it is a species of unequal proportion, according to Boethius (Arithm. i, 23). Therefore in God there is not triplicity, but Trinity.
Reply to Objection 4. In the divine Trinity is to be understood both number and the persons numbered. So when we say, "Trinity in Unity," we do not place number in the unity of the essence, as if we meant three times one; but we place the Persons numbered in the unity of nature; as the "supposita" of a nature are said to exist in that nature. On the other hand, we say "Unity in Trinity"; meaning that the nature is in its "supposita."
Reply to Objection 5. When we say, "Trinity is trine," by reason of the number implied, we signify the multiplication of that number by itself; since the word trine imports a distinction in the "supposita" of which it is spoken. Therefore it cannot be said that the Trinity is trine; otherwise it follows that, if the Trinity be trine, there would be three "supposita" of the Trinity; as when we say, "God is trine," it follows that there are three "supposita" of the Godhead.
Source: Summa Theologica (New Advent)