Objection 1. It would seem that there are not several persons in God. For person is "the individual substance of a rational nature." If then there are several persons in God, there must be several substances; which appears to be heretical.
Objection 2. Further, Plurality of absolute properties does not make a distinction of persons, either in God, or in ourselves. Much less therefore is this effected by a plurality of relations. But in God there is no plurality but of relations (28, 3). Therefore there cannot be several persons in God.
Objection 3. Further, Boethius says of God (De Trin. i), that "this is truly one which has no number." But plurality implies number. Therefore there are not several persons in God.
Objection 4. Further, where number is, there is whole and part. Thus, if in God there exist a number of persons, there must be whole and part in God; which is inconsistent with the divine simplicity.
On the contrary, Athanasius says: "One is the person of the Father, another of the Son, another of the Holy Ghost." Therefore the Father, and the Son, and the Holy Ghost are several persons.
I answer that, It follows from what precedes that there are several persons in God. For it was shown above (Question 29, Article 4) that this word "person" signifies in God a relation as subsisting in the divine nature. It was also established (28, 1) that there are several real relations in God; and hence it follows that there are also several realities subsistent in the divine nature; which means that there are several persons in God.
Reply to Objection 1. The definition of "person" includes "substance," not as meaning the essence, but the "suppositum" which is made clear by the addition of the term "individual." To signify the substance thus understood, the Greeks use the name "hypostasis." So, as we say, "Three persons," they say "Three hypostases." We are not, however, accustomed to say Three substances, lest we be understood to mean three essences or natures, by reason of the equivocal signification of the term.
Reply to Objection 2. The absolute properties in God, such as goodness and wisdom, are not mutually opposed; and hence, neither are they really distinguished from each other. Therefore, although they subsist, nevertheless they are not several subsistent realities--that is, several persons. But the absolute properties in creatures do not subsist, although they are really distinguished from each other, as whiteness and sweetness; on the other hand, the relative properties in God subsist, and are really distinguished from each other (28, 3). Hence the plurality of persons in God.
Reply to Objection 3. The supreme unity and simplicity of God exclude every kind of plurality of absolute things, but not plurality of relations. Because relations are predicated relatively, and thus the relations do not import composition in that of which they are predicated, as Boethius teaches in the same book.
Reply to Objection 4. Number is twofold, simple or absolute, as two and three and four; and number as existing in things numbered, as two men and two horses. So, if number in God is taken absolutely or abstractedly, there is nothing to prevent whole and part from being in Him, and thus number in Him is only in our way of understanding; forasmuch as number regarded apart from things numbered exists only in the intellect. But if number be taken as it is in the things numbered, in that sense as existing in creatures, one is part of two, and two of three, as one man is part of two men, and two of three; but this does not apply to God, because the Father is of the same magnitude as the whole Trinity, as we shall show further on (42, 1 and 4).
Source: Summa Theologica (New Advent)