8. How the Early Christian Theology of Arithmetic Shaped Neo-Platonism and Late Antique Christianity

After the early third century, the controversy over the theology of arithmetic disappeared from the Church. The dispute need not have died down. Gnosticizing writings well into the fourth century show a continued interest in speculative number symbolism. But the only orthodox Christian responses to be found come from recherché apologies like Epiphanius’ Panarion, indebted largely to Irenaeus for its section on Valentinianism.
On the other hand, from the late second through the early fourth centuries number symbolism took on increased importance and new roles in Platonism. The Pythagorean metaphysical innovations begun in the first centuries CE and BCE entered into the Platonist tradition, and in a sequence that echoed the Christian experience: speculative number symbolism, controversy, an attempt to articulate principles, and out of that a sense of what was acceptable in the Platonist tradition.
In this chapter I provide two accounts of the afterlife of the early Christian debate on the theology of arithmetic. The first account treats the role of number symbolism in the metaphysics and exegesis of Platonist philosophy in the late third and fourth centuries. Continuing the story from chapter 2, I continue with the legacy of Plotinus, focusing on an unusual exchange between Theodore of Asine and Iamblichus. I argue that Iamblichus, an important early fourth-century philosopher, played for Platonist orthodoxy a role comparable to that of Irenaeus and Clement. The second account pertains to the orthodox Christian theology of arithmetic. I briefly summarize how the parameters and paradigms established by Irenaeus and Clement shaped later Christian and Byzantine usage.

Platonist Number Symbolismin the Third and Fourth Centuries

As pointed out in chapter 2, the new philosophical currents that emerged from the first century BCE to the first century CE sought the origins of the world in a single ἀρχή (or a “plurality-in-unity”). These neo-Pythagorean systems explored how that unity could lead to multiplicity, and postulated multiple metaphysical layers from the topmost being down to material reality. They depended upon numbers, both for terminology and for conceptual patterns. Well into the second century, this strain of neo-Pythagoreanism—a literary ideal, not a community—coexisted with traditional Platonism. Dualism, for example, continued to be championed by philosophers such as Plutarch, who saw it as the normative motif in all classical philosophy. [1] Both monistic and dualistic systems are reported by the skeptic and anti-Platonist Sextus Empiricus (ca. 160–210). [2]
The most vivid witness to the encroachment of monistic upon dualistic territory is provided by Numenius, a second-century Platonist philosopher who flourished in Apamea. Like other Platonists of his day, Numenius was firmly committed to the philosophical importance of numbers. He wrote a treatise called About Number, and he preferred a numerical to a geometrical explanation for the composition of the soul. [3] Numbers had religious importance for him as well. He envisioned a person’s return to perfective unity as starting with the mathematical disciplines, continuing with contemplation about numbers, and terminating in the final μάθημα, that of being. [4] This religious approach to numbers was interlocked with his view of the Pythagorean tradition. In his view, Pythagoras called God by the name of Monad, and matter he called Dyad. The dyad was in no way generated from the monad; it was completely inoriginate. Numenius castigates “some Pythagoreans”—here setting recent followers against the master himself—for misunderstanding the dogma, claiming that the dyad was derived from a unique monad when the monad withdrew from its nature and sojourned into the condition of a dyad, “an absurd situation, that that which was Monad should cease to be so, and that the Dyad which had no existence should come to subsist, and that thus Matter should come to be out of God, and out of unity immeasurable and limitless duality.” [5]
Despite his hostility, Numenius was not impervious to the winds of change. His signature doctrine is a three-tiered hierarchy of principles. The first principle, which he regards as an ineffable entity, participates in, but remains above, being. Opposite this is matter, the dyad. The second level of his system involves being, which is being in motion or the demiurgic Intellect. Being comes into contact with matter, and the encounter results in opposite effects. On the one hand matter becomes unified; on the other hand, the contact with matter splits the second principle, giving it a lustful component. Out of the bifurca-ted second principle comes the third, which governs the material world. [6] Thus Numenius’ system has incorporated gnosticizing overtones, by associa-ting the natural world with the lustful activity of the Demiurge. [7] And he has embraced the metaphysical number symbolism and the multiple tiers of neo-Pythagoreanism.
The final step in the transformation from Platonist dualism to monism came in the early third century, through Plotinus (205–269/70), whose writings were so deeply influential that subsequent philosophers are frequently called neo-Platonists, as distinct from the Platonists before him. Because his literary corpus is preserved much better than that of nearly any other ancient philosopher, it is easy to treat Plotinus as a thinker ex nihilo. But he was deeply influenced by philosophers of the preceding generation, particularly Numenius and Plotinus’ teacher of eleven years, an avant-garde Alexandrian named Ammonius Saccas (fl. early third c.). Echoes of Numenius’ doctrines can be seen in Plotinus’, and the unwritten doctrines of Ammonius served as the basis for the first ten years of Plotinus’ own teaching career. [8]
Number plays such a critical role in Plotinus’ philosophy that the topic consumes studies far more extensive than this one. [9] Generally speaking, he preserves the tendency among the neo-Pythagoreans to incorporate number symbolism into metaphysics. He holds to a universe consisting of four metaphysical levels. At the very top is the One or Good, which is beyond all being and beyond all predicates. Below the One is the level of Being or Intellect. And below that is the realm of Soul, followed by matter. Number features on all four of these levels, as pointed out by Svetla Slaveva-Griffin: [10]
Being Unified number
Intellect Number moving in itself
Complete Living Being Encompassing number
Beings Number unfolded outward
These four metaphysical levels, and the prominence of various levels of “one,” are reminiscent of the Pythagorean metaphysical system reported (but not explicitly embraced) by Moderatus around a century and a half earlier.
Plotinus’ monism dominates his writings, providing a conduit to his philosophical use of number. For example, he describes the One as “the maker of number. For number is not primary: the One is prior to the dyad, but the dyad is secondary and, originating from the One, has it as definer, but is itself of its own nature indefinite.” [11] There is no room for a coeval dyad, as found in Plutarch’s and Numenius’ writings. Plotinus even declines to discuss at any length the nature and disposition of the indefinite dyad, which according to Aristotle had played such a key role in Plato’s doctrines. [12] The monistic emphasis harmonizes with Plotinus’ conception and definition of number. He acknowledges, and even uses, the Aristotelian definition of number, but he holds primarily to the generative definition. [13] He is the first philosopher we know of to take the notion of the One’s unfolding into number and reverting back to the One—a notion that underlies the definitions embraced by Moderatus and Nico-machus—and extensively integrate it into a major philosophical system. But unlike his Pythagoreanizing predecessors, as well as Valentinians and other gnosticizing groups, Plotinus prefers the term ‘hen’ to ‘monad.’ When he uses ‘monad’ and its cognates, he normally does not differentiate it from ‘hen,’ and sometimes he conflates the words. [14] So although he adopted many of the philosophical elements of the Pythagorean approach to Platonism, he preferred non-Pythagorean terminology.
Gnosticizing Christians played an important role in shaping Plotinus’ philosophy of number. They were quite active in Plotinus’ circle, and he and his students devoted considerable industry to refuting their positions. [15] One of these refutations is preserved in the last quarter of Plotinus’ lengthiest work, a treatise Porphyry titled Against the Gnostics. [16] In this treatise Plotinus, referring to an unnamed “they,” attacks “their” proposition that there may be more than three principles in the universe, the sort of proposition a Valentinian of his day might have inferred or taught. He writes about this topic only briefly, concentrating instead on the gnosticizing assaults on the goodness of the Demiurge and the created world. But the problems Plotinus discerned in Christian gnosticizing groups, problems that surface at the beginning of Against the Gnostics, seem to have remained important to him, because the very next treatise he wrote was his principal treatise on the subject, On Numbers. [17] That treatise opens with the same topic that closed Against the Gnostics, namely a consideration of evil. It then turns to the question of whether multiplicity and deviation from the One—and therefore we—constitute an evil. The work moves quickly from its original impetus and on to a meditation of what numbers are and where they stand in the metaphysical order. It is one of Plotinus’ densest and most difficult treatises.
Plotinus’ discussion takes up from Aristotle’s criticisms of Plato’s doctrine of mathematics, and attempts to broker an agreement, one that would resonate with his third-century audience. [18] He distinguishes, in the realm below the One, between substantial number (οὐσιώδης ἀριθμός, described in terms of monads) and monadic number (μοναδικῶς ἀριθμός, described in terms of henads). The former exists in the realm of being; the latter, in multiplicity. Of the two classes, he is most interested in the constitution of higher, substantial numbers, which are arrayed yet further to describe the different tiers of the intelligible realm. Being corresponds to unified number; beings, to unfolded number; intellect to number moving in itself; and complete living being to encompassing number. These four levels of number correspond to the unfolding of geometrical shape, from point to line to circle to sphere—that is, a foursome similar to Moderatus’ first system, and reminiscent of the neo-Pythagorean use of the τετρακτύς. [19]
One of the main points Plotinus argues for echoes his anti-“gnostic” theme: there is beauty and majesty even in the realm of plurality. That is, the realm beyond the One is not all decrepit. This need not be interpreted as a rebuke of gnostics such as the Valentinians; it could be a clarification, or even an affirmation of their insight. Whatever his disposition to gnosticizing Christian number symbolism, it catalyzed his own constructive explanation for where number sits in his metaphysical edifice. The highest levels of plurality in the early Christian theology of arithmetic were scions and roots of harmony. The Valentinians had idealized the Pleroma as a place of beauty (barring Wisdom’s error). Plotinus’ philosophy of number explores the rational explanation for this metaphor.
Through this treatise Plotinus strengthened in the Platonist tradition a reliance upon number in all sorts of ways. His literary executor, Porphyry (234–ca. 305), used number symbolism to arrange Plotinus’ oeuvre. [20] And Amelius (b. 216–226, d. ca. 290–300), a philosopher who, according to clues in the Life of Plotinus, may have had greater respect from the master than Porphyry had, repeatedly developed new ideas about the soul and its relationship to number. [21] He reportedly wrote a forty-book response to the Sethian treatise Zostrianos. [22] One of Amelius’ innovations was to use the numerical patterns laid out in Plato’s Timaeus as a template for classifying the various kinds of souls in the world. [23]
This trend took a new turn in Theodore of Asine (ca. 275–ca. 360), a philosopher who perhaps met Amelius in the 290s and studied with Porphyry in the opening years of the fourth century. [24] He found his inspiration in the doctrines of Numenius and Amelius, and garnered a following of at least modest size. Toward the end of Theodore’s life, his followers were still active, arguing on behalf of their master. Of his writings we know only two titles, On Names and That the Soul is All the Forms (sc. of Life), extracts of which, some lengthy, are preserved by Proclus. [25]
Theodore’s metaphysical system, as obscure today as it ever was, involves among other things a complex explanation of the origin and structure of the soul, the same theme that Amelius developed. But he bases his metaphysical edifice upon a foundation of letters, numbers, and psephic exercises previously unknown in the Platonic tradition. He explains the soul on the basis of letters, their numerical value, and the symbolism behind the numbers. The bulk of this numerically oriented system is preserved in testimony 6, worth quoting in full. [26] Proclus, our source for the testimony, says:
Theodore, the philosopher of Asine, inspired by the discourses of Nume-nius, in a rather novel manner composed treatises concerning the gene-ration of the soul, making his attempts from letters, written characters, and numbers. So that, then, we might have concisely his written opinions, come, let us make an overview, point by point, of everything he says.
[1] [27] So then, the first is rightly hymned as being, for him, ineffable, inexpressible, fount of all things, and cause of goodness. [2] And after this [first], so exalted above all things, there is a triad that defines, for him, the intelligible plane. [The triad] he calls the One [ΤΟ ЄΝ] since it comes [2a] from a breathing that is somehow ineffable (which the rough breathing of Єν mimics), [2b] from the loop of the Є itself, on its own without the consonant, and [2c] from even the Ν itself. [3] Another triad after that one defines the intellectual depth, [4] and another one, the demiurgical. The former is existence prior to being, thinking before mind, living before life. After these is the demiurgical triad, containing first, being; second, mind; and third, the fount of souls.
[5] From that triad is another triad: absolute soul, universal [soul], and [soul] of the all. We have earlier discussed the distinction of these things, each of which proceeds from the entire demiurgical triad—that is, one from being, another from mind, and the other from the fontal soul. [28] Indeed, it was proposed that Plato speak about this soul of the all, especially about the plain soul that comes from the fount of souls, about the universal [soul] and the [soul] of the all, and about the fount itself. For all things exist in all things, even if at one time in one way, and at another time, another:
in the soul before the triad [all things exist in all things] in unity;
in the plain [soul], [all things exist in all things] in wholeness before the parts;
in the universal soul, [all things exist in all things] in wholeness from the parts;
and in the third [soul], [all things exist in all things] in [wholeness] in the parts.
[All this is said] on the basis that Plato classified all these things and needed to refer to every [soul] every ratio, ratios that allow no difference among them.
And [Theodore] thinks it necessary to say first why [the soul] exists from three means. And indeed he says that the soul as a whole is a geometrical ratio, existing from both the first god according to being and the second [god] according to mind. For these very things are two essences, one undivided and the other divided. Both the arithmetical ratio (which bears the image of the first essence) and the harmonic [ratio] ([which bears an image of the] second [essence]) result in [the geometrical ratio]. The former is monadic, since it is without extension; the latter is discrete, but harmonically so.
[7] Then, the entire number might be a certain geometrical number, since [the soul] is shown to be a tetrad, being from the tetrad of the elements/letters. But lest you suspect that this number is lifeless, taking for the third heptad the first, you will find life in the extreme letters.
Rather, setting out according to its order the base of the first letter, you will see the soul is an intellectual life. E.g. Ζ, Ο, Ψ. The middle [term, Ο,] is the circle, being the intellectual one, since mind is the cause of the soul. The smallest [term, Ζ,] shows [that the soul is] geometrical, a kind of mind, through the attachment of the parallels and the diameter. [The mind] remains above and encompasses opposites and is shown to be a form of life, both oblique and not oblique. The largest [term, Ψ,] is the element/letter of a sphere. So then the lines, bent into each other, will make the sphere. On top of this, the next letter’s bases, Δ, Μ, Υ, are simultaneously three and tetradic. And because of this, as they beget the dodecad, they result in the twelve spheres of the all. The largest of the bases[, Υ,] shows that its essence yearns for two certain things and stretch up toward two affairs. Therefore some call this letter ‘philosopher.’ The [essence] of both flows to the lower [region]. So this is why we find the Υ referred to by some of the noteworthy [authors of the past]. It is between two spheres, the Ψ and the Χ [i.e. in the word ΨΥΧΗ], the former being warmer (because of the breath) and more life-giving, and the latter having each [quality] to a lesser extent. Thus, there is again a mean between two minds, one earlier and the other later, and the middle character makes clear its property and relation to the other. Rather, even though the letter Ψ is a sphere, Plato assigned the Χ to the soul, so that it might show the equal balance of motion itself, since all the lines in the Χ are equal, and thus to make the automation of the soul evident. But if the Demiurge brings in the soul through existence itself, it is clear that he himself has ordered it in proportion to the Χ. After all, that is the foremost mind. And so, because of these things, he says that the soul, as it advances and brings itself out, is a certain middle essence of two minds. And this is the manner in which these things are to be understood. But through the last letter, the Η, the advance of [the soul] up to the cube is to be observed.
[8] And if it is a dyad because of the otherness of life, and it is a triad because of the tripartition of its essence, then it has, on the one hand, the ratio 3:2. But as it enters into itself and, through its entrance, applies the dyad to the triad, it begets the hexad. As it connects to the undivided and to the trisected the harmony that [comes] through these things in doubleness, it comes into existence. And since, on the one hand, the triad, as it turns into itself, results in the ennead, and the dyad, on the other hand, moving into itself dyadically always results in the octet, so from both it results in the ratio 9:8.
The linear birth [of the soul] makes clear its indivisibility, and its thorough homogeneity (after all, every part of a line is a line), and that all the ratios are everywhere. But the split into two demonstrates that its form is dyadic. And its indivisible wholeness is an image of the first mind, whereas the unsplittable [wholeness] of the two (which he calls the circle of the same) [is an image] of the second [mind]. And the [wholeness] split into six [is an image] of the third [mind], the last to be calculated. And the octet becomes manifest from the dyad of the soul, whereas the heptad depicts in monads the first form of the soul; in decads, its intelligible [form] (because of the circle); and in hundreds, the soulish mark, the third one remaining. And [the soul’s] straight, connate nature exists for the fixed [sphere], which begets; whereas the exit and indefinability [exist for] the wandering [sphere]; and the return after the advance [exists for] the life that wanders without wandering. And since on the other hand the shape of the soul is like Χ, and its form is dyadic (since the split is in two), and the dyad [applied] to the hexad (being primarily the base of Χ) creates the dodecad, you might take from that the first twelve ancient souls.
Although the testimony continues, it is worth commenting on a number of ideas that chaotically emerge. Theodore’s system reaches a new level of complexity, similar to the intricacies of Marcus’ theology. It usually takes several slow readings to understand much of Theodore’s system. Proclus’ summary is terse, and even in faithful translations many ideas are incomprehensible. [29]
At the top of Theodore’s system (marked in the translation by [1]) is a transcendent entity, described in negative terms that describe its role as cause and source. Significantly, he does not call it “One.” Rather, it is absolute, beyond names, beyond even spirit or breath. [30]
Below this transcendent entity is the intelligible plane [2], consisting of a triad that unfolds and manifests the primal entity in the letters of ἕν. The rough breathing [2a], which Theodore claims is silent, mimics the ineffable breathing of the uppermost entity. [31] This ineffable breathing is symbolized by the rough breathing, the δασεῖα, which had not been pronounced for quite some time. [32] The unspoken δασεῖα is the perfect symbol of the ineffable, since it embodies the paradox of unspoken speech. According to another of Theodore’s testimonies, its combination with the Є, shaped by a loop [2b] (remember, Theodore is commenting on uncial Greek letters, not minuscule) renders the vowel at once utterable and unutterable. The Ν [2c] rounds out the image of the ineffable made effable. [33] The triad  ЄΝ consists of two groups: a dyad (presumably ЄΝ) and, above it, a monad (presumably ). [34]
Beneath this level is the intellectual depth [3], consisting of a triad: existence, thinking, and living. Each of these three parts of the intellectual depth stands over the elements in the triad just below, the demiurgical depth [4], a triad that consists of being, mind, and the fount of souls. This fourth level seems to have been an easy target for Theodore’s critics, who saw this doctrine, taken or derived from Amelius, as unnecessarily introducing three demiurges, in lieu of Plato’s one. [35] The threefold demiurgical level, however, sustains Theodore’s commitment to a metaphysics of triadic descent. It also critically explains the next level down, the realm of the soul [5], which consists of three kinds, all derived from the demiurgical triad in general, and the fount of souls in particular. [36] Each of the three aspects of the soul corresponds to a certain relationship between the whole and the parts, and to the three major kinds of ratios. What corresponds to what is treated in yet another testimony, where Theodore goes to some lengths to correlate the numbers one, two, three, four, and six with the four elements, and by extension with the soul. [37] For example, the series one, two, four, a geometric series (b/a = c/b), corresponds to Earth because of its name, and so the number seven, the sum of the series, also represents the Earth.
Theodore’s complex number symbolism, linking souls to the sublunary region, shows that he saw numbers as a key constituent in the formation, design, and explication of the universe. The most basic parts of this number symbolism are evident in the structures of his metaphysics, presented in testimony 6 [1–5], which may be diagrammed:
  τὸ πρῶτον ἄρρητον ‘the ineffable, first’  
νοητὸν πλάτος‘intelligible breadth’
νοερὸν βάθος‘intellectual depth’ εἶναι‘existence’ νοεῖν‘thought’ ζῆν‘life’
δημιουργικὸν βάθος‘demiurgical depth’ ὄν‘being’ νοῦς‘mind’ ζωή, πηγὴ τῶν ψυχῶν, πηγαῖα ψυχή‘life’, ‘fount of souls’, ‘fontal soul’
  αὐτοψυχή‘absolute soul’ ἡ καθόλου ψυχή‘universal soul’ ἡ τοῦ παντὸς ψυχή‘soul of the All’
Here is vividly portrayed Theodore’s mathematically molded view of the divine realm. [38] The highest realm of the universe is divided into thirteen places: four planes, each composed of a triad, all governed by a transcendent thirteenth entity. Theodore’s system is unusual, but not unique. Other Platonist texts, such as the Nag Hammadi text Marsanes, contemporary perhaps with Amelius, agree in the number of levels and in the presence of an ineffable, transcendant entity located in the thirteenth place. [39]
Later in testimony 6, Theodore points to the soul as consisting of four elements or letters [7]—the play on the double meaning of στοιχεῖον is evident—as the reason for calling the entire number of the soul a geometrical number. This introduces a letter-by-letter, numerical explanation of the word ΨΥΧΗ, which is geometrical since it consists of four letters. Theodore’s method here focuses on the number of letters in a word, not their psephic value. [40] He takes a letter, for example Ψ, finds its base (πυθμήν), then lists other letters that share the same base, in this case, Ζ and Ο. [41] That is, Ψ, Ο, and Ζ have values of 700, 70, and 7. He then uses each of the three letters, particularly its shape, to explain the letter as a whole. The combination of parallel and oblique lines in the letterform Ζ explains how the soul preserves both kinds of directions, as specified in Plato’s Timaeus. The circle of the Ο represents the mind’s generation of the soul. The ’s crossarms, arching in toward each other, represent a sphere. For the Υ in ΨΥΧΗ, its base of four, when multiplied by the three letters that share it as a base, yields the “twelve spheres of the all,” presumably the zodiac. Its shape too illustrates the philosophical implications of the letter, as its stem and arms portray the flow from upper regions to lower, and the return. The two arms of the Υ stretch toward its companion letters, Ψ and Χ, the former being more life-giving (as Theodore stipulates elsewhere, Ψ recalls Ζ, the initial of ΖΩΗ ‘life’), the latter more soullike. [42]
The final part of the testimony [8] deals with the Timaeus’ double and triple progressions and identifies the activities of soul with the various numbers that make up the most basic musical intervals (e.g. 3:2 and 9:8, the perfect fifth and the whole tone). [43] The passage is too complex to explore here. Nevertheless, note how deeply Theodore’s explanation of the constitution of the world soul involves itself in arithmetic. Theodore is squarely in the center of Pythagorean Platonism, which identified number, combined with motion and the mixture of same and other, as the source of the soul’s origin. [44] Theodore indulges in this tradition, and expands the application of numbers, making arithmetic a constituent part of every level of his philosophical system. He introduces letter symbolism to the interpretation of the Timaeus. In this he resembles Monoïmus, who took the shape of the ἰῶτα as a reflection of the divine. He has even more in common with Marcus, who introduced letter symbolism into Valentinian protology. Marcus, for example, identifies the number of letters in the name ‘Jesus Christ’ as a symbol of the structure of the aeonic realm; so Theodore treats the four letters of ψυχή as cosmologically significant. Theodore finds the shape of the letters symbolizing important metaphysical truths, just as Marcus locates in the divisions of the alphabet signs of Pleromatic emanations. Both indulge in rudimentary psephy, revealing a common assumption that words, names, and numbers have an interconnected significance well beyond what is immediately apparent. Analyzing the psephic value of a word’s letters allows one to discover hidden knowledge, and to learn more about the universe or about the person or thing that bears the name.
Just as Marcus’ speculation became a prime target for Irenaeus, so Theo-dore’s earned him the reproach of Iamblichus (ca. 245–ca. 326), his older contemporary and rival Platonic philosopher. Iamblichus, quite possibly a relative or descendant of Monoïmos, [45] had also studied with Porphyry. He later established a philosophical circle in Syria, one that Theodore eventually joined as his student. [46] Iamblichus’ criticism is preserved in the last part of Theodore’s testimony 6. Proclus says: [47]
Thus Theodore philosophizes these kinds of things about these matters, making his interpretations out of letters and utterances (to compare a few [ideas] among many). But the divine Iamblichus lambasted this sort of viewpoint in his responses, Against the Circle of Amelius (for so he titles the chapter) and indeed also [in Against the Circle of] Numenius. [Iamblichus] either—for I cannot say [which]—identified [Theodore] with these [two men] or had somewhere found them writing similar things about these matters.
So the divine Iamblichus says first that you shouldn’t make the soul the entire number or the geometrical number because of the quantity of its letters. For ΣΩΜΑ [‘body’] too is made of the same [number] of letters, as is even ΜΗ ΟΝ [‘nonbeing’]. So [by Theodore’s reasoning] ΜΗ ΟΝ would be the entire number. You could find many other things that consist of the same [number] of letters yet are shameful and completely opposite to each other, all of which would certainly not be right to conflate and confound with each other.
Second, it is dangerous to try [to build a system] based on written characters. After all, these things are relative: the carving of an archaic [character] used to be one way, but is now another. For instance, the Ζ upon which that man builds his argument had neither parallels that were completely opposed, nor the middle diagonal bar. Rather, [the crossbar was] perpendicular, as is apparent from ancient stelae.
Third, to reduce [numbers] to their bases and to preoccupy oneself with them, [going] from one number to another or vice versa, alters our understanding. For the heptad in monads, the [heptad] in decads, and the [heptad] in hundreds are not the same thing. So if this [heptad] was in the term ΨΥΧΗ, why must he sneak in an account about bases? After all, using this same technique we might transform every thing into every number, by dividing or adding or multiplying.
So much for the general [problems]. [Iamblichus] also refutes [Theodore’s] individual results, [showing them to be] fraudulent and insane. And everyone who would like to know the weakness of every point may easily acquire the treatise and read through the appropriate refutations of everything [taken] from [Theodore’s] writings.
We see in this passage a quote from a quote from a quote, and it may be helpful to sort out what is before us. Proclus, our source for testimony 6, had before him a work by Iamblichus, a book of polemic divided into chapters discussing and arguing against a series of philosophical schools. [48] One chapter of that book was directed against the circle of Amelius, and another against that of Numenius. Proclus himself is in the dark about what he is reading. He is uncertain whether Iamblichus meant to identify Theodore with these two circles, or merely to refute an approach shared by all. [49] This suggests to me that Iamblichus did not name Theodore specifically in his polemical work, but refuted Theodore’s doctrine under the name of these two schools, leaving the reader to make the connection. [50] In any case Proclus, who knew Theodore’s writings well, recognized that Iamblichus’ critique applied to Theodore.
Iamblichus makes three criticisms of Theodore’s method in general, and then refutes some specific doctrines. Proclus reports only the general criticisms. The first is Iamblichus’ contention that one cannot infer from the premise “ ‘soul’ consists of four letters” that the soul is the entire number, or that it is the geometrical number. This first criticism deals with Theodore’s fascination with the number of letters in ΨΥΧΗ. His critique, Iamblichus says, has nothing to do with the practice of gematria, but with letter counting. [51] He demonstrates the faulty logic by deploying a reductio ad absurdum. Both ‘body’ and ‘non-being’ have four letters, so Theodore’s method should allow one to declare a similar, exalted position for either corporeality or nonexistence. Iamblichus notes that a number of four-letter words, too rude to mention (he would have appreciated our English idiom), could be inserted in Theodore’s system. Would that make them just as exalted as ‘soul’?
The second criticism is that letterforms are a faulty starting point because of their relative nature. Iamblichus focuses on how letterforms change, demonstrating it with the Ζ. He points out that the archaic form of the letter was I, a shape that undermines Theodore’s interpretation of the soul’s being at once parallel and oblique.
Third, Iamblichus criticizes as valueless the practice of taking alphabetic numerals, finding their base value, and extrapolating from that base to other numerals with the same base. He questions why bases should even be a point of consideration. The risk in the practice is that someone could take a word like ‘soul’ and derive any preconceived result, through arbitrary mathematical operations. Iamblichus may be somewhat unfair here to Theodore, who follows a principle common in psephic games, that one may move from one number to another by dividing or multiplying by ten. But Iamblichus correctly criticizes the practice for its attempt to achieve preconceived results. The technique could be used to prove whatever one desired.
Now, Iamblichus was no enemy of number symbolism. His writings are full of Pythagorean arithmology. And he was the first Platonist to use number symbolism to organize his own literary output. His magnum opus, On Pythago-reanism, occupies ten books, to pay homage to the perfection of the number ten. [52] And each of the extant books in this series deals in some way with number symbolism, if not arithmetic. In the third book, Common Mathematical Knowledge, for instance, Iamblichus argues for a theory of mathematics. He defends Pythagorean mathematics as useful for sharpening one’s understanding and for purifying the soul in preparation for union with the divine. [53] His metaphysics was built on mathematics just as Plotinus’ was, and he had a system of triads that resembled Theodore’s (a triadology characteristic of subsequent Neoplatonism). [54] Iamblichus’ numerical interests were so well known that he was credited with a handbook of number symbolism popular throughout the Byzantine period, the otherwise anonymous Theology of Arithmetic. [55]
Like Theodore, Iamblichus engages in extensive symbolic analysis of the numbers in the Timaeus. Both men explain the generation of the soul using arithmetical explanations for the generation of the dyad and triad from the monad. [56] So Iamblichus criticizes Theodore, not for number symbolism per se, nor even for extravagance (that being a matter of taste). His main complaint is that Theodore’s method is capricious, bound to changeable human convention, and liable to lead one to all sorts of absurd conclusions. Theodore’s method destroys the very essence of a symbol. Iamblichus prizes numbers as symbols because of their inherent connection with higher realms of reality. Theodore focuses, however, on facets of numbers and letters that are rooted in the human condition, the product of changing social conventions. In these contexts the symbol loses its power because it depends upon temporal realities, not eternal ones. To locate numerical symbols in the realm of the divine was completely justified and expected in the Pythagorean and Platonic traditions. But to root them in human convention, as Theodore did, was not.
The debate between Iamblichus and Theodore compares favorably with that between Irenaeus and Marcus. Both Iamblichus and Irenaeus deploy biting sarcasm in their reductiones ad absurdum, to discredit an approach that threatens to disregard the rules for attaining truth. Both writers criticize techniques that are custom-bound and apt to lead a person to preconceived ideas, or to absurd, immoral positions. Both Irenaeus and Iamblichus participate in communities grounded in tradition, and they both work with systems and texts that were considered to have roots, so to speak, in higher, divine soil. For Irenaeus, this is the apostolic rule of faith, deposited in the churches and codified in the Bible; for Iamblichus it is the Platonic rule of faith, as practiced in religious theurgy and enshrined in the writings of Plato. The logic had similar effects in both traditions: just as, after the third century, number symbolism was excluded from Trinitarian theology (see below), so in the Platonist tradition the use of gematria and letter symbolism to explicate the highest metaphysical levels ended with Theodore’s followers. Neo-Platonists followed their orthodox Christian counterparts.
The comparison should not be pressed too far. Christians and Platonists had incommensurate notions of community and tradition. Christians identified themselves as the new Israel, the holy people of God, and they were committed to the Bible, the inspired charter of their faith. Platonists, who had no church to speak of, were intellectual journeymen, maintaining a tradition of discussing the issues discussed by Plato, Aristotle, and others. Nevertheless, both Irenaeus and Iamblichus characterize their opponents as having strayed beyond the acceptable limits of a tradition both sacred and true, and apply nearly identical logic in their critiques. Their targets, Marcus and Theodore respectively, also share a common approach and set of assumptions about the importance of numbers and letters, regarding them as symbols and portals to the divine.
This is not to suggest that Iamblichus was reading Irenaeus and Theodore was reading Marcus. Rather, Christianity filtered the conceptual air that the Platonists breathed. The theological issue about the appropriate use of numbers first arose in Christianity. The trends and tensions discernible in texts written by the Valentinians, Irenaeus, Clement, Monoïmos, and the Paraphrase of the “Apophasis Megale” migrated to the Platonist tradition. First, Plotinus responded to the number symbolism of the “Gnostics” by offering a thorough, penetrating defense of the metaphysical priority of numbers. His successors combined that emphasis with long-standing mathematical interests of ancient philosophers to formulate a triadology that dominated Platonist philosophy all the way through Damaskios and the closing of the academy at Athens (529 CE). [57] This later tradition built upon Iamblichus’ theology of arithmetic, but excluded Theodore’s psephic techniques.

The Later Christian Tradition

And what of that Christian experience? Where did it end up during the late third and early fourth centuries? To answer this, it is important to recall some of the main points already covered.
The Pythagoreanism that emerged in the first century BCE inspired and even provided the intellectual foundation for the theological systems of the Valentinians, the Barbelo-Gnostics, deutero-Simon, and Monoïmus. All these groups make Pythagorean number symbolism a central part of their Christian theology. They begin with an arithmetical array of aeons, multiple beings initially projected by a monad or monad–dyad pair that is the source of all things. From this top tier down to the physical world, various descending levels of reality are projected. The arithmetical edifice and terminology of the Middle Platonists helped the Valentinians describe the emanations of the aeons, their cascade into multiplicity, and the formation of the natural world and the human being. They brought to these metaphysical systems an emphasis on salvation, telling a story of how we can be or have been rescued from the present world and restored to that metaphysical edifice. Many aeons are given numerical names drawn from Pythagorean lore (e.g. τετρακτύς, ‘monad,’ and ‘dyad’) and placed in arithmetical relationships to one another, making explicit the mathematics already implied by the Pleromatic structure. Some systems are philosophically pure, committed thoroughly to monism or to dualism. That is, some focus on the utter solitude of the Monad; others make the eternal relationship between Monad and Dyad central; other systems seem not to have cared. All the models, however, fall somewhere along a monadic–dyadic spectrum. Whether pure or mixed, the different gnosticizing visions all reflected the intellectual concerns of Middle Platonism, in which monism and dualism were live, competing options. This arithmetical theology first developed in the 160s and lasted through the mid-third century. Marcus, who in the 180s fused his theology of arithmetic with grammar and letter symbolism, marks the apogee of that era, an era that coincides, as Christoph Markschies has observed, with a period of highly philosophical and well-developed gnosis. [58]
Middle Platonists also inspired early Christians to incorporate number symbolism into their interpretive techniques. Christians had always been attuned to symbolism and allegory. The exegesis of the Hebrew scriptures developed by various New Testament authors is similar to the interpretive approaches of Philo, Plutarch, and other Middle Platonists. But with the Valentinians came a surge of interest in unlocking the secrets behind numbers found in the Bible. They even used the technique to interpret the natural world. Everyone—Valentinians, Marcus, Monoïmus, Pythagoïmus, deutero-Simon, Irenaeus, and Clement—read the numbers found in Scripture in light of Greco-Roman number symbolism—the kind espoused by Plutarch and the Theology of Arithmetic—to press home their exegetical points. Those numbers could appear in the Bible explicitly, as with the 99 in the parable of the lost sheep, or it could be merely implied, as in the story of the Transfiguration, which itemizes but never numbers those present. Or it could be even more cleverly hidden, in the psephic values of letters or alphabetic numerals, such as the 801 of the word ‘dove,’ or the numeral 10 in the story of Gideon or the iota not missing from the Law. Christian exegetes latched onto the numbers they read and interpreted them in the light of some other part of their tradition. This impulse developed among orthodox and heretics alike, but in different measures, and to different ends. Clement typifies later, Byzantine number symbolism, for he more than any orthodox predecessor creatively interpreted the number symbolism of the Bible, yet he channeled it to serve orthodoxy, to ensure that the numbers discovered in the Scriptures maintained the traditions of the entire Church, not the private fantasy of an elite group. In this he was more consistent than Irenaeus, who made use of the very exegetical techniques he faulted.
The two interrelated approaches to number symbolism—the philosophical and the exegetical—met different fates. The orthodox theology of arithmetic that emerged in the mid-third century rejected the philosophical, metaphysical use of number, at least one that did not start with the rule of truth. But it permitted an exegete to be as fanciful or reserved as he liked when it came to interpreting the numbers of the Bible.
The first of these two, the demise of number symbolism in descriptions of the godhead, is not difficult to explain. All the systems discussed in chapters 3 through 5 above come off as a form of polytheism. The Valentinians and others like them were telling a story, fundamentally, of how one or two original divine principles begat a pantheon of individual divine entities. These could not be interpreted as mere metaphors, the arbitrary symbols of modern semiotics. They were to be treated as real, and as meaningful as the material world to which they eventually led. This paradigm stood at odds with the monotheism of Irenaeus, Clement, and the church traditions they defended. Irenaeus is unambiguous in his commitment to God as three persons, Father, Son, and Holy Spirit, all separate from Creation. For Clement, the one God remains free of any mathematical models, metaphors, or constraints, including that of the One. For both, neither God nor the Christian tradition is subject to numbers, but rather the converse.
This is not to say that the orthodox were unified in how arithmetic should be used. Irenaeus writes often about one Father and one Son. The oneness he emphasizes steers clear of philosophical metaphors. Clement is more willing to indulge in them. For him, God transcends any predicate, including that of the One. Clement considers numerical analogies fruitful, and he is willing to think about God as a One that stands above whatever else there is below it (including the conceptual hierarchy Monad → One) until the metaphor runs out of steam. He is comfortable with such dispensable analogies, whereas Irenaeus avoids them. The two approaches mark the range of approaches used by Christian theologians throughout late antiquity. Mathematics rarely features in the later Fathers’ trinitarian theology, and where it does it is left as an undeveloped simile—one of many—for a God who defied metaphor. Orthodox Christians deployed Platonist terminology or philosophical metaphor subversively, both to draw pagans out of philosophical attachments and into the Church, and to deflect criticism from fellow churchmen for Hellenizing the faith.
The development of number symbolism in later Christian exegesis is also not difficult to explain. Just as Irenaeus and Clement handled numerical symbols differently, in later Christianity there was no single acceptable way to use number symbolism. Those inclined to mystical, speculative, or allegorical theology tended to embrace Clement’s pattern; those more skeptical tended toward Irenaeus’, or omitted it altogether. The prolific Biblical exegete and theologian Origen (185–ca. 255) frequently uses number symbolism in his interpretation of the Bible, in a fashion somewhat between that of Irenaeus and Clement of Alexandria. A third-century bishop of Laodicea, Anatolius, who may have been the teacher of Iamblichus, collected Pythagorean numerical lore into a handbook, parts of which are preserved in the Theology of Arithmetic. [59] His interest and skill in mathematics were fundamental to his rationale in explaining the date of Easter, a logic that proved influential in the fourth century and beyond. [60] The Bible commentaries of Didymus the Blind, Evagrius of Pontus, Jerome, Augustine, and many more are salted throughout with creative number symbolism. Some of their contemporaries, such as Athanasius and John Chrysostom, were not so inclined. But this difference did not lead to controversy or dispute, since all agreed that number symbolism was to be drawn from the tradition, not imposed upon it. This left latitude for those so inclined to creatively develop number symbolism, and throughout the medieval period local traditions flourished in the Latin West, in the Byzantine East, and in the Syriac, Coptic, Ethiopic, Arabic, Georgian, and Armenian traditions. Every one of these regional traditions fostered new kinds of arithmology, collectively indebted to the Greco-Roman tradition and the formative second- and third-century arguments.
kalvesmaki fig7

Figure 7. The Transfiguration of Christ. Göreme, Karanlık Kilise, 11th c. (Photo courtesy Mustafa K. Turgut.)

A vivid example of the influence of the early Christian theology of arithmetic appears in the Byzantine icon of the Transfiguration. From the earliest, sixth-century examples into the modern period, Orthodox Christians have depicted Christ atop mount Thabor, flanked by Moses and Elijah, standing above the three apostles. Together the six figures form a symmetrical circle or triangle, to emphasize their number and harmony. Behind Christ is a mandorla of light. In the earliest example, a sixth-century fresco from Mount Sinai, the mandorla emits eight beams of light. That eight-rayed motif is found frequently thereafter. In the Palaiologan period the mandorla was arranged as two superimposed rhombuses or squares, their eight corners prominently arrayed (see Figure 7). The eight divine rays are no accident, nor is the human hexad, marking the mountain, as a triangle. Clement’s orthodox interpretation of the number symbolism behind the Transfiguration has become indelibly imprinted in the Orthodox tradition, furnishing for this icon number symbols that proclaim Christ as both man and God, the ‘episemon ogdoad.’ [61]


[ back ] 1. Opsomer 2007:381, 389–390, 395.
[ back ] 2. Sextus Empiricus Against the Logicians 10.276–283 (dualistic), 10.281 (monistic); Outlines of Pyrrho-nism 3.153–154 (monistic).
[ back ] 3. Frags. 1c, 39.
[ back ] 4. Fragment 2.
[ back ] 5. Sed non nullos Pythagoreos uim sententiae non recte assecutos putasse dici etiam illam indeterminatam et immensam duitatem ab unica singularitate institutam recedente a natura sua singularitate et in duitatis habitum migrante—non recte, ut quae erat singularitas esse desineret, quae non erat duitas subsisteret, atque ex deo silua et ex singularitate immensa et indeterminata duitas conuerteretur (fragment 52, trans. Dillon 1996:373).
[ back ] 6. Fragment 11; Dillon 2007:366–374.
[ back ] 7. Previously noted at Dillon 2007:369.
[ back ] 8. Numenius: Dillon 1996:366, 372, 374, 381. Ammonius Saccas: Porphyry Life of Plotinus 3.
[ back ] 9. See Nikulin 2002 and Slaveva-Griffin 2009.
[ back ] 10. 2009:87.
[ back ] 11. ὁ τὸν ἀριθμὸν ποιῶν. Ὁ γὰρ ἀριθμὸς οὐ πρῶτος· καὶ γὰρ πρὸ δυάδος τὸ ἕν, δεύτερον δὲ δυὰς καὶ παρὰ τοῦ ἑνὸς γεγενημένη ἐκεῖνο ὁριστὴν ἔχει, αὕτη δὲ ἀόριστον παρ’ αὐτῆς· (Plotinus Ennead–7, trans. Armstrong 1984).
[ back ] 12. Ennead 5.1.5 and 6.7.8 are, to my mind, the only clear examples. Plotinus uses δυάς mostly of duality in general, as a correlate term to μονάς, τριάς, and δεκάς, to explore sets of multiple items.
[ back ] 13. Plotinus Ennead 6.3.12, Slaveva-Griffin 2009:49–50.
[ back ] 14. Ennead–18. See also Nikulin 2002:77; Slaveva-Griffin 2009:92–94.
[ back ] 15. Porphyry Life of Plotinus 16.
[ back ] 16. Ennead 2.9. On the kinds of gnosticizing Christians Plotinus was refuting, see Turner 2006:26–27.
[ back ] 17. Ennead 6.6; Porphyry Life of Plotinus 5.
[ back ] 18. For Plotinus’ philosophy of number, beyond the outlines sketched in this paragraph, see Slaveva-Griffin 2009:85–99, 122–130.
[ back ] 19. Griffin 2009:122–126. See also pp. 52–58 above, and p. 185 below.
[ back ] 20. See Slaveva-Griffin 2009:132–140.
[ back ] 21. On respect: Amelius kept Plotinus’ notebooks, he was given authority by Plotinus to refute Porphyry, he was too busy to correct copies of Plotinus’ lectures that Porphyry requested, and he was listed by Longinus as one of the two chief Platonists teaching in Rome (the other being Plotinus): Porphyry Life of Plotinus 1–5, 7, 16–21. On Amelius in general, see Brisson 1987.
[ back ] 22. See Turner 2006:26–27, esp. n. 19.
[ back ] 23. Stobaeus Anthology–24,–4 and Brisson 1987:838–839.
[ back ] 24. Brisson 1987:818–819.
[ back ] 25. On Theodore’s life see Brisson 2002; PRE 5 (n.s.): 1833–1838 (Theodore, no. 35); Gersh 1978:289–304.
[ back ] 26. Theodore’s fragments are collected in Deuse 1973. Testimony 6 comes from Proclus Commentary on the “Timaeus,” in Festugière 1966–1968:2.274–278. For the Greek text, see the Appendix, p. 199.
[ back ] 27. My numeration and lettering appears in square brackets to mark passages for discussion below.
[ back ] 28. This refers to a discussion preserved in test. 22.
[ back ] 29. For other translations, of varying degrees of accuracy, see Proclus Commentary on the “Timaeus,” trans. Festugière 1968:3.318–321; Ferwerda 1982:51–52; Funk et al. 2000:214–216. Taylor, the great nineteenth-century English Platonist, skipped it, explaining, “Proclus gives an epitome of this theory, but as it would be very difficult to render it intelligible to the English reader, and as in the opinion of Iamblichus, the whole of it is artificial, and contains nothing sane, I have omitted to translate it” (Taylor 1820:141n1).
[ back ] 30. Pace Funk et al. 2000:214, 216, 227. It is inaccurate to call it the One because in test. 9 (Deuse), Theodore explicitly denies that this highest entity has a name.
[ back ] 31. Test. 9: sicut et spiritus tacitus and spiritum … indicibilem.
[ back ] 32. Gignac 1976:134–135.
[ back ] 33. Test. 9 is unclear. See Morrow and Dillon 1987:590 and n. 113. Turner’s suggestion (Funk et al. 2000:216) that ἝΝ represents point, line, and plane reads too much into the passage.
[ back ] 34. Test. 9 neither stipulates which letters are the monad and which the dyad, nor elaborates on how the monad generates the dyad.
[ back ] 35. See test. 12, translations by Taylor 1820:260; Funk et al. 2000:218–219; Festugière 1968:4.165–166.
[ back ] 36. For amplification on this level, see test. 22, which is translated in Funk et al. 2000:221–223, Festugière 1967:3.262–265, and Taylor 1820:92–94.
[ back ] 37. Test. 22. See n. 36 above.
[ back ] 38. See comparative schemes at Deuse 1973:22–24, Funk et al. 2000:230, Finamore 2000:256–257, and Edwards 2006:122–124.
[ back ] 39. But prominent differences should also be noted. Marsanes’ lowest realm is the corporeal world, not the soulish. And even though Marsanes teaches that the twelve stages culminating in the thirteenth consist of four levels of triads, those triads do not follow the consistent pattern found in Theodore: existence-thought-life. The regularity of Theodore’s triads lends itself to depiction in a rectangle, where both horizontal and vertical relationships are important. Theodore, for instance, places living over life (the two terms are cognates), but their corresponding entities in Marsanes, self-generated power/triple perfect and super-(corporeal?) do not have this kind of vertical relationship. Marsanes’ thirteen seals are presented seriatim.
[ back ] 40. See below for Iamblichus’ criticism, which confirms that Theodore’s method is about letter counting, not psephy. Sambursky 1978 suggests that the term ‘geometrical’ in test. 6 preserves the earliest use of the term ‘gematria.’ But what very basic psephy there is has nothing to do with what Theodore calls the “geometrical number,” a term that refers to the concept of geometrical ratio, discussed just before.
[ back ] 41. Theodore’s use of πυθμήν to refer not merely to the base but to its multiples of ten and one hundred goes against normal use of the word. Ordinarily Ο and Ψ would have a single πυθμήν, Ζ, and would not themselves be πυθμένες. See LSJ, s.v., and the psephic numerology attested in Hippolytus Refutation of All Heresies 4.13–14.
[ back ] 42. On the Υ as a Pythagorean symbol of moral choice see Harms 1975 and Dornseiff 1925:24.
[ back ] 43. Plato Timaeus 35b: 1, 2, 4, 8 and 1, 3, 9, 27.
[ back ] 44. This interpretive tradition of the Timaeus began with Xenocrates. See Plutarch Genesis of the Soul in the “Timaeus” 2, 22.
[ back ] 45. See Dillon 1987:865 for analysis of a medieval report (Photius Biblioteca 181 [126a]) that Iamblichus had ancestry from Sampsigeramos (fl. 60 BCE) and Monimos.
[ back ] 46. Eunapius Lives of the Sophists 458. On Iamblichus’ life, see Dillon 1987:863–875.
[ back ] 47. In addition to those listed in n. 29 above, other translations are in Dillon 1973:165–167 and Taylor 1820:141.
[ back ] 48. Here I agree with Festugière against Dillon (1973:337), who understands the text to refer to one title (Against the Circle of Amelius and Numenius), not two (as in my translation). The two chapters and their titles, described as refutations, show that the source is probably not a commentary on the Timaeus, as Dillon surmises, but a critical summary of various philosophical schools of thought.
[ back ] 49. My interpretation depends upon taking the τοῦτον of Festugière 1966–1968:2.277.30 with Theodore, and the ἐκείνους of the same line with the circles of Amelius and Numenius. Dillon’s reading (1973:165, 337–338), which assigns τοῦτον to Numenius and ἐκείνους to the circle of Amelius, is possible. But the overall context of Proclus’ report puts Theodore front and center, much closer (and therefore τοῦτον) than Amelius and Numenius (here only obliquely referred to through treatise titles, and so ἐκείνους).
[ back ] 50. See Dillon 1973:338.
[ back ] 51. Contra Dillon 1973:338–339 and Sambursky (see n. 40 above).
[ back ] 52. For the structure of this work, see O’Meara 1989. On the perfection of ten, see p. 54n78 above. For examples of Iamblichus’ symbolism beyond those given here, see Shaw 1999.
[ back ] 53. Iamblichus Common Mathematical Knowledge 18.7–23.
[ back ] 54. Dillon 1987:883, 888–889.
[ back ] 55. No internal evidence associates the Theology of Arithmetic with Iamblichus, but the substance of the treatise, with all its theological and scientific number symbolism, fits well with his general outlook. Tarán 1981:291–298 argues that the text, dated traditionally to the fourth century, is probably the product of a later Byzantine compiler who drew from arithmological treatises of Nicomachus of Gerasa, Anatolius of Laodicea, Iamblichus, and others. O’Meara 1989 overturns some but not all of Tarán’s assumptions. The date and textual history of the Theology of Arithmetic needs to be revisited.
[ back ] 56. Proclus’ Commentary on the “Timaeus” depends considerably on Iamblichus. See Dillon 1973:160–163, 322–325.
[ back ] 57. Wallis 1992, O’Meara 1989.
[ back ] 58. Markschies 2003:85–100.
[ back ] 59. On Anatolius, see Dillon 1987:866–867.
[ back ] 60. Knorr 1993:184; Anatolius On the Computation of the Pasch.
[ back ] 61. Not all icons show eight rays, but it was certainly the dominant tradition. See Miziołek 1990 and Drpić 2008:241–242. Clement’s Transfiguration number symbolism was known in later Byzantium: see Gregory Palamas Sermon 34.4–6 (PG 151:425–428).