Karl Barth proposed that the Trinitarian formula of "one God in three persons" be updated to "one God in three modes of being" (or "... ways of being"). Is Barth teaching Sabellian modalism? No! The reason for the change, is that Barth believed that the word "person" has substantially changed in meaning to include an "attribute of self-consciousness" (especially in the Post-Reformation era). Barth therefore rejects "three persons" because it communicates that there are three "personalities" (or self-consciousnesses) in God, and this is the heresy of tritheism!
In this post, I will explain why Barth's updated formulation of "one God in three modes of being" is a better expression of the trinitarian teachings of the early Church Fathers in their battles with Anti-Trinitarian heresies, and how "modes of being" more faithfully translates the early church creeds into modern vernacular than the outmoded term "persons".
Modes of Being (Seinweise)
Barth selects the German word "seinweise", translated as "modes of being" or "ways of being", to replace the word "person" because it is more faithful to early Church father's description of the Trinity, than modern meaning of "person". Barth explains that "seinweise" is not a completely new term, because it is a literal translation of the ancient concept of "modus entitativus" (or τρόπος ὑπάρξεως) used by the early Church Fathers to describe the Trinity.
Hence we are not introducing a new concept but simply putting in the centre an auxiliary concept which has been used from the very beginning and with great emphasis in the analysis of the concept of person. . . . God is One in three ways of being, Father, Son and Holy Ghost . . . "Mode (or way) of being" (Seinsweise) is the literal translation of the concept τρόπος ὑπάρξεως or modus entitativus as, e.g., Quenstedt [...] put it in Latin. —Karl Barth 
The phrase "modes of being" (seinweise) sounds scary at first, because of initial term 'mode'; yet, this is not a novel Barthianism! I assure you, despite the similar phonetics, that Barth is not affirming Sabellianism or Modalism whatsoever. Barth explains that "modes of being" is not a novellum, but repeating ancient church expressions that faithful describe the trinity, and do not introduce an attribute of self-consciousness that "persons" has wrongly done. (Barth provides many examples of quotations from Church fathers that use the term "modus" in small print sections in loc.)
. . . by preference we do not use the term "person" but rather "mode (or way) of being," our intention being to express by this term, not absolutely, but relatively better and more simply and clearly the same thing as is meant by "person."—Karl Barth 
And for my Biblicist readers, who desire a prooftext, Barth believes that "modes of being" is biblically justified by in Hebrews 1:3 (e.g. "the impress of His subsistence") as follows:
Heb. 1:3 already called the Son χαρακτὴρ τῆς ὑποστάσεως θεοῦ i.e., in His mode of being an "impress" or countertype of the mode of being of God the "Father." —Karl Barth 
Barth's rejection of "Persons"
Barth explained that "person" in the 19th century came to mean "personality" (or a thinking thing pace. Descartes), which had unintended result of altering the meaning of "three persons" to now communicates that there are "three personalities" (or three self-consciousnesses) in God, which Barth rejects as tritheism heresy.
What is called "personality" in the conceptual vocabulary of the 19th century is distinguished from the patristic and mediaeval persona by the addition of the attribute of self-consciousness. This really complicates the whole issue. —Karl Barth 
Barth argues that "person" originally meant "persona", as it was first used by the early Church fathers in their fight against Anti-Trinitarian heresies, and by the early credal definitions of the Trinity; for instance, the Athanasian Creed describes god as "three personas" (lit. lat. "tres personae"). Barth believed that "personas" was superior to "persons" because "personas" correctly describes God as a threefold-I. Barth rejects old formula "three persons" because there are not "three-I's in God".
"Person" as used in the Church doctrine of the Trinity bears no direct relation to personality. The meaning of the doctrine is not, then, that there are three personalities in God. This would be the worst and most extreme expression of tritheism, against which we must be on guard at this stage. . . . But in it we are speaking not of three divine I's, but thrice of the one divine I. —Karl Barth 
Due to the confusion shrouding Latin group of words (such as person, persona, personality, etc.), Barth chose the phrase "three modes of being" following the Greek group of words (such as hypostasis, substantia, substance, etc.) to replace of the deprecated "three persons".
We have avoided the term "person" in the thesis at the head of the present section. It was never adequately clarified when first introduced into the Church's vocabulary, nor did the interpretation which it was later given and which prevailed in mediaeval and post-Reformation Scholasticism as a whole really bring this clarification, nor has the injection of the modern concept of personality into the debate achieved anything but fresh confusion. The situation would be hopeless if it were our task here to say what is really meant by "person" in the doctrine of the Trinity. —Karl Barth 
Person (Augustine) is a compromise between Persona (Latin) and Hypostasis (Greek)
The phrase "three person" derives from the Athanasian Creed (or Symbol of Quicunque Virt, c. 6th century CE), which says: "And in this Trinity none is before, or after another; none is greater, or less than another. But the whole three Persons (tres personae) are coeternal, and coequal." (The Athanasian Creed is a gloss on Augustine's On the Trinity, not Athanasius.)
Barth believed that "person" was a term popularized by St. Augustine as a compromise between the Western Churches (that used the Latin word "personae") and the Eastern Churches (that used "hypostasis") in their respective definitions of the Trinity: The Latin word for person ("personae") sounded like a modalist heresy to the Eastern Churches, and the Greek word for person ("hypostasis") sounded like a tritheism heresy to the Western Churches. In On the Trinity, Augustine declared that neither term was suitable, and proposed that the term "person" would be used, rather than to say nothing at all. Augustine is arguably the greatest of the Latin Church Fathers, so the Western Latin term "person" became ubiquitous.
Karl Barth picks up this ancient feud between the Western and Eastern Churches, and argues that the Eastern Greek term "hypostasis" was the superior term, because the Western Church's Latin term persona implied modalism. Barth argued that persona (πρόσωπον) describes the Trinity like three masks (that are hiding the face of a fourth unknown God). Consequently, Barth's term "modes of being" is a conscious decision to set aside all the Western derivatives of "person" and side with the Eastern Churches vernacular of "hypostasis", because history has shown that the Church erred by using "person" instead of "hypostasis".
The word persona, πρόσωπον like trinitas, which is supposed to have been used first by Tertullian, originates with the controversy against the Sabellian heresy and is thus designed to denote the being in and for themselves of Father, Son and Spirit respectively. But did not persona, πρόσωπον, also mean "mask"? Might not the term give new support to the Sabellian idea of three mere manifestations behind which stood a hidden fourth? In view of this the Greek Church largely preferred to translate persona by ὑπόστασις rather than πρόσωπον. On the other hand ὑπόστασις necessarily suggested to the Westerners substantia in the sense of natura or essentia. and so they saw themselves threatened here by the proximity of tritheistic ideas. Finally, if the West clung to persona and the East to ὑπόστασις neither party could be perfectly content with the other nor ultimately with itself.
It is something of a relief that a man of Augustine's standing openly declared (De trin., V, 9, VII, 4) that to call what is meant "person" is simply a necessitas or consuetudo loquendi. A really suitable term for it just does not exist. —Karl Barth 
Karl Barth's proposal of "one God in three modes of being" explains the Trinity much better than the traditional formula of "one God in three persons". However, Barth's replacement of "persons" with "modes of being" has caused most English speakers to think Barth is advocating for Sabellian modalism. So the received meaning of "modes of being" or "ways of being" also causes misunderstanding.
Barth is correct that "person" has had an "attribute of self-consciousness" added to it. However, the meaning of "person" is also symbolically linked to the trinity, such that when the "threefold-I of God" is described, the term mostly commonly used is "person". So it becomes impossible to separate "person" from the threeness of God.
Also, there's great danger in substituting "modes of being" for "person", because the injection of "and the son (filioque)" to the Nicene Creed, is linked to the Great Schism, and the Church has had enough schism. So in my opinion, I'm happy to continue to use the term "person" or "three persons", so long as it is immediately explained to mean "three modes of being" and not "three personalities" as in tritheism.
Perhaps the most famous critic of Barth's trinitarian theology is Jürgen Moltmann. In Moltmann' Trinity and the Kingdom of God, Moltmann said that Barth's reformulation was a victory for Sabellianism. Moltmann didn't explicitly say Barth was a Sabellian, but that his theology may be exploited to assist Sabellianism. Moltmann advances a theology of the trinity that is much further in the direction of tritheism, than Barth, and I understand this criticism from Moltmann as Moltmann asserting his difference from Barth. Only people who hate Barth would so lazy and careless to assert that Moltmann believed Barth was a heretic—that's just nonsense.
"But viewed theologically this is a late triumph for the Sabellian modalism which the early church condemned. The result would be to transfer the subjectivity of action to a deity concealed 'behind' the three Persons." —Jürgen Moltmann 
Karl Barth's criticism of the Trinitarian formula "one God in three persons" is correct because "person" no longer adequately conveys the same meaning it did when it was used by the early Church fathers and creeds. After the Post-Reformation era, an attribute of self-consciousness was added to the meaning of "person" that had the unintended consequence of changing the meaning of "three persons" to the tritheistic "three personalities". The term "person" was always provisional, and understood to be an insufficient compromise between the Greek speaking early Eastern Churches who preferred the term "hypostasis" and the Latin speaking Western Churches who preferred the term "personae". Barth rightly blames the great Latin Church Father, St. Augustine of Hippo, for influencing the Church to adopt the term "person" as a compromise. Barth believes that the Church erred by following the Western Church's use of "person", and believes that a return to the Eastern Church formulate better explains the Trinity, especially now that the meaning of "person" has changed. Barth's proposal of "one God in three modes of being" is a more faithful rendering of the Trinitarian formulas of the early Church, and more clearly explains the Trinity than the traditional "one God in three persons".
Karl Rahner and others, have made the same criticisms of "person" and "three persons", so Barth's argument is convincing and well supported, so I encourage anyone who prefers to follow Barth, to do so. Moltmann and other critics overstate their argument when they claim that Barth's proposal of "one God in three modes of being" is a victory for Sabellian modalism. The primary danger of Barth's proposal, is the changing of traditional phrases so late in Church history, may cause schism in the Church. Then again, theology is never finished, and must constantly be translated into modern vernacular. If the old formulate of "one God in three persons" is retained, then there's an immediate need to explain this formula a "one God in three modes of being".
[^Header Image Background] By Andrei Rublev - From here., Public Domain, Link
[^1] Barth, Karl. Church Dogmatics, Vol 1.1 Doctrine of the Word of God. Vol. 2. London: T & T Clark, 2010. Print. Study Edition. [359-60].
[^2] Ibid. .
[^3] Ibid. .
[^4] Ibid. .
[^5] Ibid. .
[^6] Ibid. .
[^8] Moltmann, Jürgen. The Trinity and the Kingdom: The Doctrine of God (Systematic Theology Contributions). Minneapolis, MN: Fortress, 1993. 139. Print.