The Paraconsistent Numbers and the Set Theory Implied in the Cappadocian Trinitarian Doctrin

Lourié, Basil

Institute of Philosophy of Saint-Petersburg State University

The Paraconsistent Numbers and the Set Theory  Implied in the Cappadocian Trinitarian Doctrine Basil Lourié, National Research University Higher School of Economics (HSE) hieromonk@gmail.com An intuition of pseudo-natural numbers. The Cappadocian doctrine of Trinity is intrinsically paraconsistent. Several Byzantine authors have discussed its paraconsistent features explicitly. For instance, Evagrius Ponticus (345–399) statedμ “The numerical triad is followed with the tetrad, but the Holy Trinity is not followed with the tetrad. It is not, therefore, a numerical triad. The numerical triad is preceded with the dyad, but the Holy Trinity is not preceded with the dyad. Thus, it is not a numerical triad” (Gnostic Chapters VI, 11-12). From any modern set-theoretical viewpoint, the main problem of the “set theory” and the very idea of number possibly implied in such statements is the complete absence of the pairs. Indeed, there were several less Orthodox interpretations of Trinity where some kind of pairing was provided, but the Byzantine mainstream throughout the whole Byzantinische Jahrtausend was following the path specified by Evagrius (and, before him, his teacher Gregory of Nazianzus). The absolute prohibition of any kind of pairing results into inapplicability of the axiom of Pairing (and, therefore, the axiom of Infinity either) and, what is the most important, any standard set-theoretical definition/interpretation of the notion of number such as that of von Neumann (based on the notion of ordered pair). Of course, this means that, in the framework of von Neumann’s approach (and taking the side of Couturat and Zermelo in their discussion with Poincaré), the Trinity is not the number three and even not a number at all. However, this conclusion sounds counterintuitive. And, indeed, in the framework of Poincaré’s understanding of number as an “intuition du nombre pur” [1μ122], there is no problem in acknowledging that the root “Tri-” in “Trinity” has something to do with the number three… or, at least, some kind of number three. Our present task is to define what exactly kind of numbers is meant. In the presently available studies in paraconsistent mathematics there is no description of such mathematical objects [2–3]. I would propose in advance to call the kind of numbers we are looking for “pseudo-natural numbers”. Ternary exclusive OR and the notion of pseudo-ordered pair. The Trinity does not allow pairing because it is not governed with the ordinary exclusive disjunction. The latter regulates the choice between such propositions as “this hypostasis is the Father” or “that hypostasis is the Father”. If there are three hypostases, this choice is to be repeated. The table of truth-values of the function corresponding to the ordinary exclusive disjunction shows that this connective, even though preventing any two hypostases from being both the Father, allows being the Father to the three hypostases simultaneously. Somewhere in the Christian Orient such a conclusion was accepted—but not in Byzantine Patristics. To exclude this possibility, we have to use a quite different connective, the so-called ternary exclusive OR (⊻3), where the choice is presumed to be performed directly from the three and without the choices within the pairs at all. This connective has been at first described by Emil Priest in 1941 but remained almost unstudied until recently [4]. The ternary exclusive OR, however, does not allow constructing the pairs. This, in turn, prevents our numbers in the Trinity from forming any kind of row, that is, an ordered consequence. From the mainstream Byzantine viewpoint (and unlike, among others, the different Western Filioque doctrines), there are not, in the Trinity, “the first”, “the second”, and “the third”. An early formulation of this statement is articulated by Severianus of Gabala in the early 5th century, but then, such authors as Nicephorus Blemmydes (1250s) and especially Gregory Palamas (1330s) and Joseph Bryennios (early 15th cent.) have further elaborated on it in the context of the polemics against the Filioque. 56The famous formula of Gregory of Nazianzus “the monad having from the beginning moved to the dyad stayed at the triad” (Sermon 29:2) was interpreted by a part of Byzantine authors (including Maximus the Confessor) epistemologically—as applied to our understanding of the Trinity but not to the Trinity per se, whereas another part of them (including, most probably, Gregory himself) understood it ontologically—as pertinent to the Trinity per se. However, the monad, the dyad, and the triad of this formula in its ontological reading were never identified with specific hypostases. This scheme of the movement of the monad was absolutely symmetrical vis-à-vis the three hypostases. We can ask, however, what means “dyad” in this scheme, if the Trinity does not allow pairing? I would name the implied logical object the pseudo-ordered pair.  The pseudo-ordered pair is to be defined as a paraconsistent generalisation of Kuratowski’s standard definition of the ordered pair. In a set of n elements, there is one element chosen to be the first, a; the remaining elements (designed with the letter b with an appropriate index) are in amount of n–1. Thus, the pseudo-ordered pair is (ܽ, ⋀ ܾ�−1�−1 ) =: ⋀{{ܽ}, {ܽ, ܾ�−1}}�−1   The above formula describes a paraconsistent conjunction: it does not design n–1 pairs, but only a unique pair with n–1 “second” elements.  For the case of the Trinity, n = 3. In the contexts irrelevant to the ȝοȞαρχία of the Father (the doctrine of the Father as the unique beginning and “the source” of the Trinity), each element (hypostasis) of the Trinity is to be chosen as the first. In the contexts of ȝοȞαρχία, where only the Father is the first (e.g., in the Filioque polemics), this formula states that the Spirit is not the third after the Son who is the second, but both are equally “the second ones”. The pseudo-natural numbers we were looking for are formed by the pseudo-ordered pairs when all possible choices of the first element are made simultaneously—in an accordance with Severian of Gabala saying that “the divine nature does not have an order—not as disordered but as being beyond any order” (Ps.-John Chrysostom, Hom. in Gen. 24:2, ch. 2). Note: Without being infinite (or “transfinite”), the pseudo-natural numbers imply a difference between their ordinality and cardinality. According to the formal definition above, the ordinality—or rather the “pseudo-ordinality”—is n, whereas the cardinality is 1. To increase the cardinality, we need to allow plurality of the first elements of the pseudo-ordered pair, a, that means allowing plurality of the pairs. This is inapplicable to the Trinity due to the principle of the ȝοȞαρχία of the Father. The plurality of a’s is not paraconsistent. Thus, the cardinality corresponds to the consistent constituent of the pseudo-natural number (the number of the pseudo-ordered pairs involved), whereas its pseudo-ordinality is paraconsistent. To encompass the doctrine of uncreated divine energies, the above definition of the pseudo-natural numbers must be generalised over the transfinite objects (that I would call “pseudo-ordinal transfinite numbers”). The present study is a part of a larger project Nr. 16-18-10202, “History of the Logical and Philosophical Ideas in Byzantine Philosophy and Theology”, implemented with a financial support of the Russian Science Foundation.  References 1. H. Poincaré. Du rôle de l’intuition et de la logique en mathématiques // Compte-rendu du deuxième Congrès international des mathématiciens — Paris 1900. Paris: Gautier-Villars, 1902, 115–130 2. Ch. Mortensen. Inconsistent Mathematics. Mathematics and its applications, 312; Dordrecht: Kluwer Academic Publ., 1995 3. G. Priest. Mathematical Pluralism // Logic Journal of the IGPL 21, 2013, 4–13 4. F. J. Pelletier, A. Hartline. Ternary Exclusive Or // Logic Journal of the IGPL 16, 2008, 75–83 57

The Paraconsistent Numbers and the Set Theory Implied in the Cappadocian Trinitarian Doctrine

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The Paraconsistent Numbers and the Set Theory Implied in the Cappadocian Trinitarian Doctrine

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Institute of Philosophy of Saint-Petersburg State University