In the Tradition ‘person,’ or in the Greek, hypostasis (ὑπόστασις) is appealed to when referring to the Father, Son, and Holy Spirit. For some this poses a problem; for Barth in fact. There are a variety of reasons why Barth does not prefer to use this word for describing the Father, Son, and Holy Spirit; primary of which is the context he found himself in, and 
the way person had come to be understood under existential pressures. Not to mention the fact that even early on in the Patristic development the language itself also was less than amenable for many of the theologians. Nevertheless, this is the language that came to be used when referring to the ‘persons’ of the Godhead. Barth opted for his ‘mode of being’ language in place of person, but most have not followed that.
There has been some concern, following von Harnack’s ‘Hellenization thesis,’ that the Greek world was uncritically imbibed by early Christians resulting in the morphing of some of the early Christian theological developments into a hybridized version of the biblical reality. Classical theists push back at this, but some like Thomas Torrance took some of that thesis to heart; in so doing, he offers an alternative account, and thus response, to the Harnackian thesis, by attempting to demonstrate that what happened wasn’t a wholesale appropriation of Hellenic philosophical conceptual patterns being imposed on Christian revelation; but instead, he argues that the early church Fathers, while taking much of the Hellenic language, retexted it in such a way that its grammar was given new conceptual orientation under the pressure of God’s Self-revelation in Christ. Peter Leithart calls this the ‘evangelization of metaphysics’;[1] at a level I agree with that. Here is how TFT reasons this through:
The basic term used in this development was the word hypostasis (ὑπόστασις) taken over from the New Testament reference to Christ the Son of God as ‘the express image of his being (χαρακτὴρ τῆς ὑποστάσεως αὐτοῦ)’. Then within the context of the Church’s deepening understanding of the Gospel the word hypostasis was adapted to express the objective self-revelation of the Son and Word of God made flesh in Jesus Christ full of grace and truth who encounters us and speaks to us face to face as the incarnate ‘I am’ of the living God. But it was only when this was further thought out in the light of the three-fold self-revelation of God as Father, Son, and Holy Spirit that the specific concept of ‘Person’ took shape, and then only within the inter-personal relations of the Holy Trinity as one Being, three Persons. In the course of this development the term hypostasis was filled out through association with ‘name’ or ὄνομα used in its concrete sense, ‘oneself’ or αὐτός and especially ‘face’ or πρόσωπον, to refer to self-subsistent self-identifying subject-being in objective relations with others. In its theological deployment, therefore, the term hypostasis was not taken over from Greek thought unchanged, but was stretched and transformed under the impact of God’s trinitarian self-revelation through Christ and in the Spirit to such an extent that it became suitable to express the identifiable self-manifestation of God in the incarnate economy of divine salvation as Father, Son, and Holy Spirit – that is, as three distinctive hypostatic Realities or Persons. This change from a Hellenistic impersonal to a Christian personal way of thinking is very evident in the penchant of early Church theologians to qualify words for ‘God’, ‘Word’, ‘being’, ‘life’, ‘authority’ etc., by attaching to them the expression for ‘oneself’ or αὐτός, as for example αὐτοθέος, αὐτολόγος, αὐτουσία, αὐτοζωή, αὐτοεξουσία, in order to stress the intensely personal nature of God’s interaction with us through the presence and power of his Word and Spirit. Thus Logos, far from being some impersonal cosmological principle, is identified with the Son or Word of God, the divine Аὐτολόγος incarnate in Jesus Christ who reveals himself to us fact to face πρόσωπον κατά πρόσωπον, and speaks to us in the Holy Scriptures directly in person, ἐκ προσώπου, or αὐτοπροσώπως.[2]
I don’t know about you, but I rather like the way Torrance reasons his way through this. What we end up with, as we always do with Torrance, is an emphasis on the personal reality of who God is. Not in a contrived, existentialist, or modern sense, but in the revealed and biblical sense of what it means to be personal as that is given in and for us in the antecedent life of God pre-destined for us in the Son of the Father by the overshadowing of the Holy Spirit.
But you see the way Torrance disentangles this issue, which frankly, is a rather knotty issue in the history, by reasoning from the inner-theo-logic itself, as that is given in the Self-revelation of God in Christ, and allowing that to be the material reality that then goes on to shape the grammar adopted from the Greeks? Some people charge Torrance with hagiographically reading the Fathers by imposing his own style of personalist reading onto what they in fact did with the Hellenic pool of grammar they were working with. Personally, whether or not he does engage in this sort of liberty, does not ultimately bother me; what we end up with, material-theologically from Torrance is where the cream is. Yes, we want to be as accurate as possible with the way we engage with the history of ideas, and the history itself. But the greatest gift a constructive theologian, par excellence, like Torrance can give us, is the reasoning he did about God’s filial life for us; for the church catholic and the world for whom He gave His life in flesh and blood.
For my money, we need to know that God loves us, and still recognize at the same time that God is God and we are not. This is what Torrance does for us in a masterful way. He honors the Creator-creature distinction, while, in an Athanasian key, emphasizing God as He is in Himself as a prius, in the eternal bond-age of Self-given love as Father of the Son, Son of the Father in the koinonial fellowship of the Holy Spirit. Christians need to know that this is the God with whom we have to do. It is in this that we can avoid the moralistic therapeuticizing deity who dominates most of the minds of the evangelical churches (in North America and the West); and at the same time walk in an intimate hand-in-hand relationship with the wonderous and ineffably triune God of eternal life who is Father, Son, and Holy Spirit.
[1] In his book on Athanasius.
[2] Thomas F. Torrance, The Christian Doctrine of God: One Being Three Persons (London: Bloomsbury T&T Clark, 2016), 156.