Kalvesmaki, Joel. 2013. The Theology of Arithmetic: Number Symbolism in Platonism and Early Christianity. Hellenic Studies Series 59. Washington, DC: Center for Hellenic Studies. http://nrs.harvard.edu/urn-3:hul.ebook:CHS_KalvesmakiJ.The_Theology_of_Arithmetic.2013.
2. Generating the World of Numbers: Pythagorean and Platonist Number Symbolism in the First Century
Two intellectual traditions from classical antiquity laid the foundation for the early Christian theology of arithmetic. The first, and most easily identified, was the rich tradition of number symbolism in the ancient Mediterranean. Numbers had been used symbolically from very early times and in many cultures, as attested by cuneiform tablets and Egyptian hieroglyphs. In the Greco-Roman world of the first century, strands of number symbolism drawn from different periods and cultures had coalesced into a single amorphous tradition. For inhabitants of the Roman Empire it provided a rich storehouse of ideas with which they could interpret omens, religious texts, the natural world, and the mathematical sciences.
The second great tradition was metaphysical speculation. The earliest Greek philosophers were noted for asking how many sources or roots—ἀρχαί—there were to the universe. Was it monadic, dyadic, polyarchic, or a synthesis? How many levels of reality, if any, existed above the material world? If the universe started out as a monadic unity, how did the second element or level originate? What kind of metaphysical entity were numbers? These questions were discussed by the earliest Greek philosophers, including Plato, but they were addressed with renewed vigor in the first century BCE, when so-called Middle Platonist philosophers, inspired by Pythagorean speculation, put forward new metaphysical theories inextricably linked to the definition and symbolism of numbers.
In this chapter I discuss these two traditions, which early Christian theology brought together. To explain the culture of number symbolism in the first century—the ways numbers were used in antiquity to interpret literature and the natural world—I turn to the writings of Philo of Alexandria and Plutarch of Chaeronaea. Both men incorporate number symbolism into their arguments, and so show us its importance in persuading ancient readers to adopt a certain perspective or interpretation. In the second half of the chapter I turn to the Greek philosophical tradition. After setting out in very broad strokes the metaphysical issues that concerned classical philosophers, I focus on the innovations of early Middle Platonism, namely the Pythagorean and Platonist philosophies described by Alexander Polyhistor, Eudorus of Alexandria, and Moderatus of Gades. I offer new interpretations of texts written by the last two, and in so doing I explain how Greek metaphysics took a turn toward monism and numerically patterned metaphysics.
Numbers to Interpret the World: Philo of Alexandria and Plutarch of Chaeronea
In the first century, the rich mélange of number symbolism that circulated in the Mediterranean was associated with Pythagoras, but that association was ana-chronistic. Pythagoras, who had flourished seven centuries earlier, left behind no writings, and his followers had disappeared by the time of Aristotle. Whatever lore they had in arithmetic, geometry, and music had been assimilated into general intellectual culture and dissociated from these extinct Pythagoreans: the pseudo-Pythagorean texts composed between the time of Aristotle and the founding of the Roman Empire contain very little mathematical symbolism. 
With Nigidius Figulus (d. 45 BCE) began a movement to recover and lionize Pythagoras and his ideals. That movement, which we call neo-Pythagoreanism,was a literary or cultural ideal that claimed, among other things, that Pytha-goras was the fountainhead of philosophy (mainly Plato’s) and of number symbolism (its actual origins notwithstanding).  As Christianity developed, neo-Pythagoreanism became more widespread. In the first century CE the philo-sopher Moderatus of Gades wrote ten or eleven books of Pythagorean lore. Around the same time, Apollonius of Tyana adopted the lifestyle and teachings of Pythagoras and become an itinerant holy man. In the second century, Nicomachus of Gerasa, Numenius of Apamea, and Theon of Smyrna all wrote mathematical and philosophical texts laced with Pythagorean lore.
The influence is seen in Philo of Alexandria (fl. early first c. CE), a Jewish writer and political leader who was called a Pythagorean in antiquity, in part because he used allegorical number symbolism extensively.  He is one of the earliest writers known to have systematically compiled a handbook of ancient number symbolism. This treatise is lost, but an idea of its contents can be gathered from many passages in Philo’s extant writings.  His arithmology is significant for this study because it shows that number symbolism pervaded Jewish culture, which Christianity then shared, as much as Greco-Roman.
A good example is found in one of Philo’s best-known texts, On the Creation, in which he argues for the intellectual coherence, even superiority, of Moses’ account of the creation. Near the beginning of the treatise Philo explains why God is said to have created the world in six days. It is not as if God, who conceived and executed everything all at once, needed the extra time. Rather, in those six days God supplied order and rank to created beings. According to the laws of the nature of numbers, six is the number most conducive to begetting. Philo explains:
For it is the first perfect number after the monad, equal to its parts and composed by them (half is a triad, a third is a dyad, a sixth is a monad). And it is, so to speak, male and female, fitted together by the power of each. For in things that exist odd is male and even is female. So the beginning of odd numbers is the triad, and of even numbers the dyad, but the power of both is the hexad. For the world, being the most perfect of created things, was put together in accordance with the perfect number, the hexad. And since it was about to have in itself things created from copulation, it had to be fashioned in accordance with a mixed number, the first even-odd. It was to encompass the form of the male, who sows the seed, and that of the female, who receives offspring. 
In this extract Philo presents several number symbols, some of which he assumes his readers know and accept. The first pertains to six. In antiquity, perfect numbers were those equal to the sum of their factors (including the number one). Hence six, whose factors are one, two, and three, is a perfect number. The second symbol pertains to the numbers two and three. All numbers were considered to have gender: even numbers were female, and odd numbers, starting with three, were male. (The number one was frequently considered androgynous.) The gendering of number was both extensive and old, going back to Aristotle and the Pythagoreans.  These assumptions explain Philo’s observation that six is the first number born from the multiplicative union of male and female numbers. So it makes sense that the creation happened in six days: no other number better typifies the fertility God implemented in the natural order. Philo builds upon both of these number symbols, one illustrating perfection and the other sexual generation, to argue that to be perfect and productive, creation had to have occurred in six days. Philo uses number, both here and throughout his writings, to unite seemingly disparate worlds. 
Philo’s number symbolism was not exceptional. Flourishing shortly after him, Plutarch of Chaeronea used it just about as frequently, and in a similar fashion. In the sixties CE, Plutarch studied with Ammonius, an Alexandrian Platonist for whom mathematics and number symbolism were central. Ammonius’ arithmo-logical influence is seen throughout Plutarch’s diverse oeuvre, especially in two treatises, now lost, devoted exclusively to numbers: Whether the Odd or Even Number Is Better and Monads.  But he shows contradictory attitudes toward number symbolism: in some passages he revels in it, and in other places he ridicules it.
This ambivalence is seen in The E at Delphi, which is styled as a Socratic or Aristotelian dialogue. Plutarch includes himself and Ammonius, his former teacher, as two of the six named participants. The subject at hand is the meaning behind an “Ε” inscribed at Delphi. The various participants note that the inscription could be read as a number or a letter, and either reading could be interpreted in different ways. One by one the participants offer seven possible explanations for why the letter was inscribed on the temple.
The first answer, proposed by Plutarch’s brother Lamprias, takes the Ε as a numeral, which he says represents the five wise men of Greece: Khilon, Thales, Solon, Bias, and Pittakos, all of whom are reported to have met at Delphi, where they agreed to consecrate the letter in honor of their number (Plutarch of Chaeronea The E at Delphi 3 [385d–386a]). Ammonius dismisses the suggestion, as does the participant who offers the next explanation, that the Ε, being the second of seven vowels, represents the second of the seven planets—the sun, which followed the moon in ancient cosmology—and therefore Apollo (4 [386a–b]). This suggestion too is dismissed.
The third and the fourth hypotheses explain the Ε (in Plutarch’s time it was called ee, not epsilon), according to the way it was spoken, ει. So one participant says that the Ε represents either the word ‘if’ (εἰ), the keyword used to discover from the god the outcome of a future endeavor; or the ‘if’ governing the optative mood, to indicate wishes or prayers (4–5 [386b–d]). Another claims that the ‘if’ indicates the force of syllogistic logic (6 [386d–387d]).
The next two participants return to the possibility that Ε is a numeral. An Athenian suggests that it symbolizes the number five, “the pemptad” (7 [387d–f]).  Plutarch eagerly takes over the idea and pursues a lengthy excursus on the mathematical and symbolic excellence of the number five. He notes that five is the sum of the first odd and even numbers, and is therefore called ‘marriage’ (8 [387f–388c]). Five is called ‘nature,’ since when multiplied by itself the product’s final digit is always five; when five is multiplied by any other number it results in either the decad or itself; and this behavior resembles nature, which returns either to itself or to perfection (8 [388c–e]). But what does this have to do with Apollo, the god at Delphi, who was traditionally associated with the numbers one and seven, not five? Plutarch offers a convoluted explanation that involves Dionysus, the harmony found in poetical measures, and finally the ratio of three to nine, seen in the relation between the creation and the conflagration (9 [388e–389c]).
As if entangled in his own obscurity, Plutarch leaves this line of thought unresolved, and reverts to his original explanation, that the number five is native to divinity because it generates either itself, like fire, or perfection, like the universe. So, he goes on, five also appears frequently in music, in the interval of the fifth (literally διὰ πέντε), one of the five basic intervals (10 [389c–f]). Furthermore, Plato affirmed there to be five worlds, Aristotle taught five elements, and there are five fundamental geometrical shapes in the Timaeus.  Plutarch associates the five senses with the five elements, and he refers to Homer, who divided the cosmos into five parts (12–13 [390b–c]). He appeals to the sequence point, line, plane, and solid, and argues for its continuation to a fifth level, the soul, which in turn naturally separates into five parts. There are five classes of living things in the world: gods, daemons, heroes, people, and beasts (13 [390c–f]).  Five is the sum of the first two squares, provided that one is willing to take the monad as a square (14 [390f–391a]). Plato posits five chief principles, causes, and categories. Therefore, Plutarch argues, the Delphic inscription was set up to anticipate Plato’s doctrine (15 [391a–d]). The crowning point in this rambling encomium—now longer than all the previous five explanations combined—is a riddle about why, when the priestess is led to the Prytaneum, five sortitions are first performed. One of the participants, Nikandros, warns that the reason should not be uttered. “Until such time as we become holy men,” Plutarch answers, smiling, “and God grants us to know the truth, this also shall be added to what may be said on behalf of the Five.” 
Ammonius, offering the seventh and final explanation, responds skeptically to Plutarch’s praise. He notes that just about any number’s praises could be sung, especially Apollo’s native number seven (17 [391e–392a]). For him, the more plausible explanation for the Ε is “you are” (εἶ). For the rest of the treatise, this seventh and final explanation deals with what it means to ascribe eternal, unchanging being to Apollo, given our own fleeting, fluctuating condition (18–21 [392a–394c]).
The length of the sixth explanation shows that Plutarch was fascinated with numbers and their symbolism. But the philosophical heft of the seventh makes it difficult to determine how seriously he regarded them. The mathematical interpretation of the Ε is the longest, but it is trumped by Ammonius’ ontological explanation. Of all the participants, Ammonius, the mathematician and Pythagorean, would be the one expected to emphasize the number symbolism of the Ε. At the end of The E at Delphi, the reader is left with no definite indication as to which explanation is most to be trusted.
In his study of Plutarch, Robert Lamberton notes how difficult it is to interpret the Delphic dialogues: “deliberate but coy self-portraits” where “Plutarch remains a very elusive presence.”  In his reading, the dialogues dramatize inquiry and keep a single, dominating explanation at arm’s length, emphasizing instead the importance of dialogue and the pursuit of truth. This idealization of the pursuit—not the attainment—of truth is symbolized by the setting in Delphi, where the oracle speaks in riddles easily misinterpreted, and where the priests may not divulge the mysteries. Plutarch, himself a Delphic priest when he wrote the dialogue, would never have revealed the secrets of the priesthood, so to search the discourse for the correct answer is a fool’s errand.  But if the arithmological explanation of the Ε is merely one installment in the search for truth, then why has Plutarch dwelt on it and at such length? What does he intend the reader to do with all this numeric lore?
One obvious answer, in light of Lamberton’s thesis, is that Plutarch wants his readers to be entertained, to have their repertoire of knowledge expanded.  Such a casual attitude toward number symbolism is illustrated best in Table Talk, a lengthy collection of festive after-dinner conversations. There, number symbolism crops up in many of these discussions with an air of sport and riddle, and the points are treated as intellectual curiosities, nothing more. But in other treatises Plutarch encourages his readers to use number symbolism to understand the world. This is best seen in Isis and Osiris, an analysis of traditional Egyptian religion and mythology.
Some of the number symbolism behind Egyptian mythology Plutarch finds ridiculous but fascinating. For example, Typhon, when hunting in the light of the moon, found and chopped up Osiris’ corpse into fourteen parts. That this dismemberment relates to the lunar phases Plutarch has no doubt.  But he goes on to report the notion that cats give birth to successively larger litters—one, two, three, all the way up to seven kittens, thereby giving birth to twenty-eight in all.  He admits that the story is far-fetched, but he finds it uncanny that cats’ eyes dilate when the moon is full. 
Most of the symbolic numbers in Isis and Osiris Plutarch treats as key interpretive tools for religious myth. For example, he claims that the Egyptians assigned numbers to gods and thereby inspired the Pythagoreans. Apollo is the monad, Artemis is the dyad, Athena is the hebdomad, and Poseidon is the first cube (eight)—familiar Greek motifs that were originally part of Egyptian religion.  Plato’s nuptial triangle, consisting of sides of lengths three, four, and five, also derives from the Egyptian myth of Isis and Osiris.  The side of length three symbolizes the male, the base of four the female, and the hypotenuse their offspring. These three sides correspond to Osiris as the source (ἀρχή), Isis as the receptacle (ὑποδοχή), and Horus as the completion (ἀποτέλεσμα). The number symbolism follows a pattern of interpretation comparable to Philo’s explanation of the six days of creation: three (Osiris) is the first odd number and is perfect; four (Isis) is a square with sides of the (first) even number two; and five (Horus) is likened to both its father (three) and mother (two). In this allegory, five symbolizes not only marital union but the perfect offspring of that union. 
Although he treats numbers anthropologically, as devices for understanding human myths of the world, Plutarch also handles them with some reverence, as portals into the realm of the gods. This harmonizes with his views on symbolism in general. When approaching the stories of the gods, Plutarch avoids the two extremes of superstition and atheism.  His middle way is to adopt a philosophi-cal and pious attitude to the various customs of the world, and thereby find the truth embedded in symbols. Plutarch says that just as the planets and elements, though named differently, are held in common, so too people may give the gods different names, but they nevertheless share the same ones.  Behind all these naming systems, behind all the various religious symbols, is a single reason (λόγος) and providence that orders and guides everything.
Number symbolism was part of that single reason and providence for both Philo and Plutarch. In the first century, to engage piously with symbols was to grapple with the truth. Number symbols could explicate religious mythology and thereby persuade readers to embrace a particular religious world view. They could provide a gossamer portal to the harmonies of the world or to the divinity that pervades it. But number symbols could also be abused or misinterpreted. They could be tools of superstition, such as in Plutarch’s numbers concerning cats. Or they could be contested, just as the six days of creation could be a sign of either the inferiority or the superiority of the God of the Bible. Numbers were important for interpreting the world, and for persuading others to embrace that interpretation.
Monism, Dualism, and Pluralism
It is well known that the earliest Greek philosophers sought a single, original principle in the world, one ἀρχή to account for the existence of all things. In the sixth century BCE the Ionians grounded this principle in the material realm. Thales identified the ἀρχή as water; Anaximander as the indefinite (τὸ ἄπειρον); Anaximenes as air. This impulse to the monistic shares an affinity with even older Orphic cosmogonies, which identify night, Khronos, or water as the sole progenitor of the universe. 
Alongside the Ionian monistic impulse was a cultural tendency toward pluralism, widespread in myth. In Hesiod, the two beings Earth and Sky are a single, commingled original entity that undergoes a split, to engender Chaos. One strain of Orphic mythology has Water paired with Matter/Earth as the beginning of all things. Pherecydes held that there were three original eternal beings: Zas, Khronos, and Khthonie, conceived of both mythologically and in cosmological abstraction.  All three of these sources date to the sixth century BCE, and all three attest to a shared sense of the origins of the universe—an original plurality full of strife and struggle, a cosmic drama that explains the conflicts found in succeeding generations, those of gods, titans, heroes, and humans.
This pluralism is also reflected in other strains of early Greek philosophy, perhaps in reaction to Ionian monism. Heraclitus, for example, saw the cosmos as a theater for the strife of opposites, a strife held in tense unity by the logos, symbolized by fire. Empedocles posited as first principles the four elements: fire, air, water, and earth—four roots of the universe whose intermingling was regulated by love and strife. Empedocles’ contemporary Anaxagoras taught that an infinite number of material elements always existed, in mixture, in a kind of dualism of mind and matter. And the atomists, most notably Democritus, held to a dualism cum pluralism: the two basic principles of the universe—the full and the void—plus the infinite number of atomic elements.  The earliest Pythagoreans were also pluralists. 
Under the influence of Parmenides, monism nearly vanished from Greek philosophy. This is not because he was a dualist, since it is unclear how many principles he taught.  But Parmenides’ central argument—that existence excludes becoming and change—challenged subsequent philosophers to provide an account of the world that satisfied both his criteria for existence and our experience of a world of change. If the one original principle truly existed, it could not change. Some other principle would be required to explain the change we observe.
In light of Parmenides’ challenge, it is not surprising that Plato espoused a kind of dualism, but he cloaks it in a language of shifting terminology and metaphor. For instance, in the Timaeus the cosmogony begins with two entities: the demiurge and the receptacle. The primal existence of both is assumed, and not explained. In Philebus, Plato (through Socrates) wrestles with the relationship between the one and the many, and again he presumes but never explains the preexistence of both principles. He offers instead a process by which two opposites come to coexist in unity. Hence the third “type” in Philebus, the combined synthesis of the definite and indefinite, the classical model of balance between opposites. The notion of two original principles appears in the Republic, where the realm of ideas is subordinated to a single principle, the good. In other dialogues the good stands in opposition to a second principle, and this second principle is itself conceived of as a pair of contrasts, for instance “the great and the small.”  This last leitmotif is developed by Aristotle, who says that Plato embraced the one and the indefinite dyad (i.e. the “great and small”) as his principles. The one is the essence of the forms, and the dyad (also called ‘great-and-small’ and likened to other contrasts) is their matter. 
Some of the names for the principles—one, many, indefinite dyad—show that Plato frequently thought of the highest, immaterial world in numerical terms. This is no surprise, because he treated the entire universe, not just its immaterial component, as a chain of numbers. According to Aristotle (Meta-physics a6), Plato’s ideas exist in an upper, incorporeal realm of forms above the sensible world, and corporeal objects exist by participating in these higher ideas. The ideas and the material world are linked by an intermediate realm of numbers, distinct from the lower, material realm because they are unchanging and eternal, and from the upper, ideal realm because they are multiply instantiated. These numbers, too, come from above, through the participation of the great-and-small with the one.  Thus numbers are a layer between the noumenal realm and the material, providing Plato’s system with three levels, not the traditional two. The position departs from classical Pythagorean number theory: the Pythagoreans claimed that numbers were corporeal, and that sense-perceptible things are, or are made of, numbers.  But for Plato, number is rooted in the upper realms, not the lower material world.
Around the first century BCE a new trend emerged in Platonist metaphysics. On the one hand classical dualism was challenged, as a group of philoso-phers began to champion a pure monism.  But where one of Plato’s positions was discarded, another was magnified. The monistic principle these new philosophers envisioned, nothing like the material ἀρχαί of the Ionian philosophers, sat atop an edifice of three or more levels of immaterial reality. And the levels related to each other primarily in arithmetical terms. So Plato’s impulse to think mathematically about the world and to stratify the world into multiple levels of reality was adopted and expanded.
This trend is central to the early Christian theology of arithmetic, so here we must slow down. Although the later effects are quite evident, the original sources are scarce, and tersely reported. We have far fewer texts of Hellenistic philosophy (treatises written after Aristotle but before Plotinus) than we do for other periods, and it is easy to allow the voluminous writings of Plutarch and Philo—who did not participate in this trend—to drown out other writers from the period who were equally prolific but not lucky enough to have their works preserved. To linger on these fragmentary sources is important, because entire metaphysical systems are condensed and summarized in only a few sentences, often in obscure Greek where every word counts.
The earliest fragment attesting to this trend is found in Alexander Poly-histor’s Successions of the Philosophers (ca. first c. BCE), which quotes from a text called Pythagorean Memoirs (itself of unknown date):
“The principle (ἀρχή) of all things is a monad. And from the monad the indefinite dyad exists, like matter for the monad, its cause. From the monad and the indefinite dyad come the numbers, and from the numbers the points. From them come lines, out of which come planar figures. From planes come solid figures, and from these, sense-perceptible bodies, from which come the four elements—fire, water, earth, air.” 
The text goes on to describe the composition of the Earth in terms reminiscent of the cosmogony in Plato’s Timaeus.
We do not have enough of Alexander’s original text—indeed, we know hardly anything about Alexander himself—to know what he made of this system. But the quotation above shows that in the first century BCE there was circulating a Pythagorean cosmology with three innovative features. First, it emphasized the monadic origin of the world: only the monad is preexistent; the dyad exists because of the monad. Second, it multiplied the number of immaterial levels—seven realms unfold, one from the other, before corporeality is achieved. And third, these seven regions are primarily mathematical—arithmetic precedes geometry, and within arithmetic monad precedes dyad, both of which precede numbers. The world is described numerically, with the monad advancing step by step to fill up the material universe. Thus, although the author of the Pythagorean Memoirs draws from motifs found in the cosmogony of the Timaeus, he develops a position that differs from classic Platonism and early Pythagoreanism—indeed from any form of ancient Greek philosophy.
Eudorus (fl. late first c. BCE), who lived shortly after Alexander, also dis-cusses the Pythagoreans. Here are his words, preserved by the sixth-century philosopher Simplicius:
“It must be stated that according to the higher account the Pythagoreans call the one the principle (ἀρχή) of all things, but according to the other account there are two principles of things that are perfected—the one and the nature opposite to it. Out of everything that can be considered according to opposites that which is noble is subordinated to the one; and that which is base, to the nature set in opposition to it. Thus, these are in no way principles, by these men’s account. For if there is one principle for these things, [but] another for those, they are not common principles for all things, as is the case with the one.… Wherefore … they say that the one is the principle of all things in a different manner, as if matter and all existent things come into existence from it. This is the upper god.” 
After a brief interlude the quote continues:
“I [Eudorus] say that those in Pythagoras’ circle abandoned the one as the principle of all things, and in a different manner brought in the two highest elements. They call these two elements by various names. For one of them is called ordered, defined, knowable, male, odd, right, light; its opposite, disordered, indefinite, unknowable, female, even, left, darkness. Thus on one hand they call the one a principle, but on the other hand [they call] the one and the indefinite dyad elements—[the] one exists recurrently, so both are principles. And clearly, one of the ones is the principle of all things, but the other one is the dyad’s opposite, what they call a monad.” 
Most scholars dealing with this passage believe that Eudorus is describing a single Pythagorean account of the principles.  In my opinion, Eudorus is critically comparing two competing systems. According to what he calls the loftier version, the Pythagoreans held the one to be the principle of all things. According to their second version there are two principles, which preside over things considered opposites. Eudorus criticizes this second account for its incompleteness. The lower pair of principles cannot properly be deemed principles, as the Pythagoreans claim, since they generate not everything but only opposites. In contrast, according to their monistic version, not just some but all matter and all things exist thanks to the one, the so-called higher god. Eudorus accuses the Pythagoreans of abandoning a monistic principle and introducing a dyadic one, and of adopting new terminology, preferring the term ‘elements’ (στοιχεῖα) to ‘principles’ (ἀρχαί). He argues that because the term ‘one’ is used in both schemes, they should both describe the same entity—the one. So if the one is a principle in the first system, it should be the same in the other. But this runs contrary to their terminology and results in a serious inconsistency: the one is given contradictory functions.
This testimony shows that both monist and dualist versions of Pythago-reanism were then circulating, and that Eudorus preferred the monist one as being truer and closer to Pythagoras’ original teaching. This is a testament to how thoroughly this new strain of Platonism was rewriting the legacy of Pythagoras and Plato. Eudorus’ prejudice is confirmed by another of his preserved fragments, in which he emends the text of Aristotle’s Metaphysics to have Plato say that the one is responsible for the existence of the forms and matter, thus sub-ordinating matter to the one.  So Eudorus considers monism to be the superior philosophy, echoing the monism found in the Pythagorean Memoirs.
Arithmetic also plays a key role in the systems Eudorus reports. The terms ‘one,’ ‘monad,’ and ‘dyad’ recur throughout, and Eudorus highlights how the one functions numerically in the metaphysical scheme. His critique is built upon the notion that, on its own, the one is capable of generating all beings; but when treated as a peer of the indefinite dyad its function is incomplete, responsible for only half of a limited set of objects. Both Eudorus and his loftier Pythagorean system prefer an account of the universe where the one is responsible for the generation of all things.
The Pythagorean themes advanced by Eudorus and the Pythagorean Memoirs are confirmed by Moderatus, who flourished in the first century CE and who, like the other two, collected and commented on Pythagorean lore.  In two key passages, Moderatus outlines two metaphysical systems, the first belonging to the Pythagoreans, the second to Platonists. Moderatus’ text, like Eudorus’, is preserved by Simplicius, but this time via Porphyry’s treatise On Matter. Moderatus’ account falls in two sections. This first is somewhat paraphrastic, with explanatory comments from either Simplicius or, more likely, Porphyry. Simplicius states:
And the first among the Greeks to have apparently held this theory about matter [that the differences from materiality to immateriality are marked by invisible, immeasurable criteria] are the Pythagoreans, and after them Plato, as Moderatus also relates. For he declares that according to the Pythagoreans, the first one is shown to be beyond existence and every essence. But the second one (which is the truly existent and object of intellection) he says is the forms. And the third (which is that of the soul) participates in the one and the forms, and the final nature [that derives] from it, which is of sense perceptibles, does not participate, but is arranged in accordance with their reflection. The matter that is in [the sense perceptibles], of nonbeing first existing in quantity, is [termed] ‘eclipse-shadow’ and [is] yet even further inferior and [derives] from it. 
This passage is well known among scholars of intellectual history. Ever since E. R. Dodds wrote his seminal 1928 article, Moderatus has been widely celebrated for providing the earliest datable version of what would become the standard metaphysical edifice of Neoplatonism, inspired by the reading of Plato’s Parmenides: a series of ones in descending tiers—from transcendence to intellection to soul and finally to sense perception.  Three levels of immaterial reality transcend the material plane, completely unattached except by imitation. The material world merely reflects the upper realms, unlike the soulish realm, which participates in the higher levels. The sense-perceptible world is but an image, a shadow of the incorporeal matter that resides above it.
This highly compressed fragment is worth reading again. Although it is a sludge of jargon, it tells a complex metaphysical story. The first half outlines the relationships of a hierarchy of three ones. The second half explains the material world, in terms both of nature and of matter, which derive from the third one. The physical world is both a decorative impression of the upper ones, and the first existential instantiation of nonexistence, the result of unity’s entering the dimension of quantity. The phrase ‘eclipse-shadow’ (σκίασμα) implies a technical term familiar to Moderatus’ audience. The term alludes to Plato’s allegory of the shadows cast by a fire on a cave wall, but more directly likens the material world to the shadow cast by the earth on the moon, implying that nonexistence (μὴ ὄν), the act of first existing (πρότως ὄν), and quantity (ποσότης) were likened, in the original metaphor, to the sun, sunlight, and earth. Like the two Pythagorean systems described earlier, Moderatus’ Pythagorean system is dominated by monism, with the superexistential one presiding over lower monistic levels and bestowing on them its unitary constitution. The metaphysical edifice is built from the blocks of arithmetic, with the four levels of the universe emerging from unity through quantity into material reality.
At the beginning of the quote above, Simplicius promises Moderatus’ account of what the Pythagoreans and Plato thought about matter. The first passage presents the Pythagorean account. Simplicius’ second passage from Moderatus follows on the heels of the first, reporting the Platonist account. This time it is a direct quotation, again via Porphyry. Note the arithmetical terms ‘unitary’ and ‘quantity’:
And these things Porphyry has written about in the second book of On Matter, quoting the [comments] of Moderatus: “ ‘When the unitary logos wanted, as Plato says somewhere [Timaeus 29d7–30a6], to have the generation of beings constituted from itself, it made room for quantity by its own privation, by depriving [quantity] of all its logoi and forms. He called this a quantity shapeless, undifferentiated, and without outline, but receptive of shape, form, differentiation, quality, and everything of this sort.’ He says, ‘Plato was likely to have predicated of this quantity a great number of names, calling it all-receptive and formless and invisible and least capable of participating in the object of intellection and scarcely grasped by spurious reasoning and everything related to these [terms].
“ ‘This very quantity,’ he says, ‘and that form that is thought of according to the privation of the unitary logos, which encompasses in itself all the logoi of beings, are paradigms of the matter belonging to bodies,’ the very [matter] that he said both the Pythagoreans and Plato called quantity; not quantity as a form, but according to privation and loosening and extension and dispersion and because of its deviation from being. Because of these things matter appears to be evil, as if fleeing the good, and is apprehended by it and is not allowed to depart from its limits, as, on one hand, the extension receives and is limited by the logos of the form-magnitude, and as, on the other hand, the dispersion is made into a form by arithmetical distinction.” Thus, matter is, by this account, nothing other than the deviation of sense-perceptible forms vis-à-vis the objects of intellection, [forms that] wander off from that realm and are dragged down toward non-existence. 
Many commentators on Moderatus presume that this second passage explains the first, when in fact they are independent. This is signaled in the preamble, and by Simplicius’ quotation style (giving only the second quote a bibliographical reference). The difference is apparent too in the content. The second passage diverges from the first, not merely summarizing but expanding upon Plato’s system and its interpretive tradition, to provide the rationale for an innovative account of the structure of the universe. Simplicius, or perhaps Porphyry, had extracted two quotes from Moderatus on the concept of quantity, and tried to relate them. But they resist such a close comparison. 
In this second passage, a distinct metaphysical system is described, from the top down. At the highest point is the unitary λόγος, an entity that possesses within itself all the forms and λόγοι of existent things—exactly how this uppermost entity is constituted and how it possesses an internal plurality of forms and λόγοι is not explained. But it desires things to come into existence, so by privation it creates a vacuum for quantity, the second-level entity.  Along with quantity is an entity called ‘form,’ whose existence is also unexplained.  These two entities become paradigms for corporeal matter, which is labeled by the Pythagoreans and Plato as “quantity.” This quantity—not to be confused with the upper, ideal quantity mentioned here and at the end of Moderatus’ Pythagorean report in the first passage—is a material entity that arises through a series of acts of estrangement. Its extension is guided and regulated by the proportionate interaction of ideal quantity (here called magnitude) and form. Its dispersion is structured by numbers. The four verbal nouns—privation, loosening, extension, and dispersion—describe four stages in the creation of corporeality, starting with void in the center and ending in the dispersed material world, a non-existence. 
The numerical language—‘unitary logos,’ ‘quantity,’ ‘logoi’ (implying mathe-matical ratios), and ‘arithmetical distinction’—runs throughout Moderatus’ Pla-tonic system, presenting the origin of the world as if it were a cascade of num-bers. The system may seem monist, but the constitution of the unitary logos is unclear. This entity has volition and, as Christian Tornau points out, rationality—which entails objects that are thought, and therefore plurality.  Within the unitary logos exist forms and λόγοι, which implies that dualism, pluralism, or even an attempt at pluralism-in-unity is at work. Thus, although Moderatus reports the monism of the Pythagoreans, it is unclear whether he was himself a monist.
Taking into account all three Middle Platonist authors—the Pythagorean Memoirs, Eudorus, and Moderatus—we see a new emphasis on metaphysics, an attempt to explicate what Plato left unsaid or unclear.  Multiple levels of reality are postulated, emanating from the top down or from the center outward in imitation of the generation of numbers. To describe the cosmogony, mathematical metaphors are essential. Against the traditional dualism of Hellenistic Platonists, a new preference for monism (or at least a monism with multiplicity) had emerged, ascribed to the rather plastic category ‘Pythagoreanism,’ and treated as if it were Pythagoras’ original teaching. Pythagoreanism was a literary edifice, a memory to be filled with lore and ideas, not a living community.
The new monism was influential. We see its attraction in Philo of Alexandria, who occasionally used it to describe the divinity, treating God and his powers as if they were part of a Middle Platonist metaphysical system.  And, because the new philosophical monism was attractive to monotheists such as Jews and Christians, we will see in the next several chapters how this new monistic metaphysics, combined with the widespread interest in number symbolism, became the foundation for the early Christian theology of arithmetic.
[ back ] 1. Thesleff 1961, 1965.
[ back ] 2. Kahn 2001:88–93.
[ back ] 3. Runia 1995.
[ back ] 4. Philo’s lost treatise On Numbers is reconstructed by Stähle 1931; a fragment preserved in Armenian is found in Terian 1984. The tradition of handbooks of number symbolism began possibly with Posidonius (ca. 135–ca. 51 BCE). See Robbins 1921.
[ back ] 5. Philo On the Creation of the World 13–14: τῶν τε γὰρ ἀπὸ μονάδος πρῶτος τέλειός ἐστιν ἰσούμενος τοῖς ἑαυτοῦ μέρεσι καὶ συμπληρούμενος ἐξ αὐτῶν, ἡμίσους μὲν τριάδος, τρίτου δὲ δυάδος, ἕκτου δὲ μονάδος, καὶ ὡς ἔπος εἰπεῖν ἄρρην τε καὶ θῆλυς εἶναι πέφυκε κἀκ τῆς ἑκατέρου δυνάμεως ἥρμοσται· ἄρρεν μὲν γὰρ ἐν τοῖς οὖσι τὸ περιττόν, τὸ δ’ ἄρτιον θῆλυ· περιττῶν μὲν οὖν ἀριθμῶν ἀρχὴ τριάς, δυὰς δ’ ἀρτίων, ἡ δ’ ἀμφοῖν δύναμις ἑξάς. ἔδει γὰρ τὸν κόσμον τελειότατον μὲν ὄντα τῶν γεγονότων κατ’ ἀριθμὸν τέλειον παγῆναι τὸν ἕξ, ἐν ἑαυτῷ δ’ ἔχειν μέλλοντα τὰς ἐκ συνδυασμοῦ γενέσεις πρὸς μικτὸν ἀριθμὸν τὸν πρῶτον ἀρτιοπέριττον τυπωθῆναι, περιέξοντα καὶ τὴν τοῦ σπείροντος ἄρρενος καὶ τὴν τοῦ ὑποδεχομένου τὰς γονὰς θήλεος ἰδέαν. In this book all translations not otherwise credited are my own.
[ back ] 6. Sources and analysis in Burkert 1972:33–34.
[ back ] 7. For other examples, see refs. in nn. 1–2 above.
[ back ] 8. Lamprias’s catalogue, nos. 74, 163.
[ back ] 9. The thought is repeated in Plutarch The Obsolescence of Oracles 36 (429d). See also Plutarch Isis and Osiris 56 (374a).
[ back ] 10. Plutarch The E at Delphi 11 (389f–390a), a meditation expanded in his Obsolescence of Oracles 32–33 (427a–428b).
[ back ] 11. The soul was normally trisected in the Platonic tradition, and, indeed, in Plutarch’s other writings. Dillon 1996:194.
[ back ] 12. The E at Delphi 16 (391d–e): “οὐκοῦν” ἔφην ἐγὼ μειδιάσας “ἄχρι οὗ τἀληθὲς ἡμῖν ὁ θεὸς ἱεροῖς γενομένοις γνῶναι παράσχῃ, προσκείσεται καὶ τοῦτο τοῖς ὑπὲρ τῆς μεμπάδος λεγομένοις.”
[ back ] 13. Lamberton 2001:5.
[ back ] 14. Lamberton 2001:26, 149, 156–158.
[ back ] 15. Dillon 1973:190.
[ back ] 16. Isis and Osiris 18 (358a), 42 (368a).
[ back ] 17. Isis and Osiris 63 (376e).
[ back ] 18. Compare Galen, who more forcefully argues against the Pythagoreans, who theorized that odd-numbered days, because they are male, induce a major change (κρῖσις) in a sick patient (De diebus decretoriis 922–924). Galen attributes this power to the moon and its rhythms.
[ back ] 19. Plutarch Isis and Osiris 10 (354–355).
[ back ] 20. Plato Republic 546b; Plutarch Isis and Osiris 56 (373f–374a).
[ back ] 21. Cf. Philo On the Contemplative Life 65–66, which claims that the number five is “the most natural” (φυσικότατος) number because it is drawn from the orthogonal triangle that is the “source of generation of the universe” (ἀρχὴ τῆς τῶν ὅλων γενέσεως). For another extended application of the orthogonal triangle to generation and copulation (but without direct reference to five as ‘marriage’), see the scholia on Homer, set D, 19:119. There, the triangle is used to explain why infants are viable in the seventh and ninth months, but not in the eighth, a common belief in the ancient world.
[ back ] 22. On Superstition and Isis and Osiris 11 (355c–d), 67 (378a). Cf. Gwyn Griffiths 1970:100–101, 291.
[ back ] 23. Plutarch Isis and Osiris 67 (377e–f).
[ back ] 24. Kirk et al. 1983:22–33.
[ back ] 25. Kirk et al. 1983:34–41, 24–26, 56–57. See also Burkert 1972:38–39.
[ back ] 26. Kirk et al. 1983:414–415.
[ back ] 27. For more on Pythagorean pluralism and its relationship to Parmenides see Burkert 1972:32–35, 48–49, and Huffman 1993:23.
[ back ] 28. Patricia Curd has challenged the long-standing consensus that Parmenides was a material monist (2005:xvii–xxix).The only known pre-Platonic monist after Parmenides was Diogenes of Apollonia (Curd 2005:131). For a more complete account of the turn from monism to pluralism via Parmenides, see Rist 1965.
[ back ] 29. Burkert 1972:17, and Ross 1924:169–171.
[ back ] 30. Outlined at Metaphysics 987b14–29, with further refs. and discussion in Burkert 1972:21–22.
[ back ] 31. Ross 1924:166–168 for references and discussion. See also Annas 1976:13–21, on the possibility that the intermediate mathematicals were introduced after Plato.
[ back ] 32. Burkert 1972:31–34. Although Aristotle misrepresents the Pythagorean tradition in key places, he seems to be accurate here. For according to testimony independent of Aristotle, the Pytha-gorean Philolaus championed not numbers but limiters and unlimiteds as his highest principles. And even these principles are corporeal, not immaterial; see Huffman 1993:37–53.
[ back ] 33. On the shift from older, dualist Pythagoreanism to the innovative monist strain see Dillon 1996:344, 373, 379; Armstrong 1992:34–41; Kahn 2001:97–99; and Thomassen 2000:3–4.
[ back ] 34. ἀρχὴν μὲν τῶν ἁπάντων μονάδα· ἐκ δὲ τῆς μονάδος ἀόριστον δυάδα ὡς ἂν ὕλην τῇ μονάδι αἰτίῳ ὄντι ὑποστῆναι· ἐκ δὲ τῆς μονάδος καὶ τῆς ἀορίστου δυάδος τοὺς ἀριθμούς· ἐκ δὲ τῶν ἀριθμῶν τὰ σημεῖα· ἐκ δὲ τούτων τὰς γραμμάς, ἐξ ὧν τὰ ἐπίπεδα σχήματα· ἐκ δὲ τῶν ἐπιπέδων τὰ στερεὰ σχήματα· ἐκ δὲ τούτων τὰ αἰσθητὰ σώματα, ὧν καὶ τὰ στοιχεῖα εἶναι τέτταρα, πῦρ, ὕδωρ, γῆν, ἀέρα· (quoted in Diogenes Laertius [fl. third c. CE] Lives of the Philosophers 8.25).
[ back ] 35. “κατὰ τὸν ἀνωτάτω λόγον φατέον τοὺς Πυθαγορικοὺς τὸ ἓν ἀρχὴν τῶν πάντων λέγειν, κατὰ δὲ τὸν δεύτερον λόγον δύο ἀρχὰς τῶν ἀποτελουμένων εἶναι, τό τε ἓν καὶ τὴν ἐναντίαν τούτῳ φύσιν. ὑποτάσσεσθαι δὲ πάντων τῶν κατὰ ἐναντίωσιν ἐπινοουμένων τὸ μὲν ἀστεῖον τῷ ἑνί, τὸ δὲ φαῦλον τῇ πρὸς τοῦτο ἐναντιουμένῃ φύσει. διὸ μηδὲ εἶναι τὸ σύνολον ταύτας ἀρχὰς κατὰ τοὺς ἄνδρας. εἰ γὰρ ἡ μὲν τῶνδε ἡ δὲ τῶνδέ ἐστιν ἀρχή, οὐκ εἰσὶ κοιναὶ πάντων ἀρχαὶ ὥσπερ τὸ ἕν.” καὶ πάλιν “διό, φησί, καὶ κατ’ ἄλλον τρόπον ἀρχὴν ἔφασαν εἶναι τῶν πάντων τὸ ἕν, ὡς ἂν καὶ τῆς ὕλης καὶ τῶν ὄντων πάντων ἐξ αὐτοῦ γεγενημένων. τοῦτο δὲ εἶναι καὶ τὸν ὑπεράνω θεόν” (Simplicius Commentary on Aristotle’s “Physics” 9.181.10–19).
[ back ] 36. “φημὶ τοίνυν τοὺς περὶ τὸν Πυθαγόραν τὸ μὲν ἓν πάντων ἀρχὴν ἀπολιπεῖν, κατ’ ἄλλον δὲ τρόπον δύο τὰ ἀνωτάτω στοιχεῖα παρεισάγειν. καλεῖν δὲ τὰ δύο ταῦτα στοιχεῖα πολλαῖς προσηγορίαις· τὸ μὲν γὰρ αὐτῶν ὀνομάζεσθαι τεταγμένον ὡρισμένον γνωστὸν ἄρρεν περιττὸν δεξιὸν φῶς, τὸ δὲ ἐναντίον τούτῳ ἄτακτον ἀόριστον ἄγνωστον θῆλυ ἀριστερὸν ἄρτιον σκότος, ὥστε ὡς μὲν ἀρχὴ τὸ ἕν, ὡς δὲ στοιχεῖα τὸ ἓν καὶ ἡ ἀόριστος δυάς, ἀρχαὶ ἄμφω ἓν ὄντα πάλιν. καὶ δῆλον ὅτι ἄλλο μέν ἐστιν ἓν ἡ ἀρχὴ τῶν πάντων, ἄλλο δὲ ἓν τὸ τῇ δυάδι ἀντικείμενον, ὃ καὶ μονάδα καλοῦσιν” (Simplicius Commentary on Aristotle’s “Physics” 9.181.22–30).
[ back ] 37. My reading of Eudorus differs from that of Rist (1962:391–393), who sees Eudorus as having muddled up a single Pythagorean system, the same system Alexander Polyhistor reports. See also Dillon 1996:115–135; Kahn 2001:97–98; Trapp 2007:351–355; Bonazzi 2007; and Staab 2009.
[ back ] 38. Rist 1962:394.
[ back ] 39. For a fuller account of Moderatus, see Dillon 1996:344–351 and Kahn 2001:105–110.
[ back ] 40. (230.34) Ταύτην δὲ περὶ τῆς ὕλης τὴν ὑπόνοιαν ἐοίκασιν ἐσχηκέναι πρῶτοι (35) μὲν τῶν Ἑλλήνων οἱ Πυθαγόρειοι, μετὰ δ’ ἐκείνους ὁ Πλάτων, ὡς καὶ Μοδέρατος ἱστορεῖ. οὗτος γὰρ κατὰ τοὺς Πυθαγορείους τὸ μὲν πρῶτον ἓν ὑπὲρ τὸ εἶναι καὶ πᾶσαν οὐσίαν ἀποφαίνεται, τὸ δὲ δεύτερον ἕν, ὅπερ ἐστὶ (231.1) τὸ ὄντως ὂν καὶ νοητὸν, τὰ εἴδη φησὶν εἶναι, τὸ δὲ τρίτον, ὅπερ ἐστὶ τὸ ψυχικόν, μετέχειν τοῦ ἑνὸς καὶ τῶν εἰδῶν, τὴν δὲ ἀπὸ τούτου τελευταίαν φύσιν τὴν τῶν αἰσθητῶν οὖσαν μηδὲ μετέχειν, ἀλλὰ κατʹ ἔμφασιν ἐκείνων κεκοσμῆσθαι, τῆς ἐν αὐτοῖς ὕλης τοῦ μὴ ὄντος πρώτως ἐν τῷ ποσῷ ὄντος οὔσης σκίασμα καὶ ἔτι μᾶλλον ὑποβεβηκυίας καὶ ἀπὸ τούτου (Simplicius Commentary on Aristotle’s “Physics” 9.230.34–231.5). In my translation I have supplied in square brackets the text to be understood from the Greek text or the context. Parts of the translation enclosed by parentheses correspond to Greek text that I regard as the explanatory comments of Simplicius or Porphyry. For earlier literature see Dörrie and Baltes 1996:477, Tornau 2001, and Staab 2009:71–76.
[ back ] 41. Dodds 1928.
[ back ] 42. (231.5) καὶ ταῦτα δὲ ὁ Πορφύριος ἐν τῷ δευτέρῳ Περὶ ὕλης τὰ τοῦ Μοδεράτου παρατιθέμενος γέγραφεν ὅτι “βουληθεὶς ὁ ἑνιαῖος λόγος, ὥς πού φησιν ὁ Πλάτων, τὴν γένεσιν ἀφ’ ἑαυτοῦ τῶν ὄντων συστήσασθαι, κατὰ στέρησιν αὑτοῦ ἐχώρησε τὴν ποσότητα πάντων αὐτὴν στερήσας τῶν αὑτοῦ λόγων καὶ εἰδ(10)ῶν. τοῦτο δὲ ποσότητα ἐκάλεσεν ἄμορφον καὶ ἀδιαίρετον καὶ ἀσχημάτιστον, ἐπιδεχομένην μέντοι μορφὴν σχῆμα διαίρεσιν ποιότητα πᾶν τὸ τοιοῦτον. ἐπὶ ταύτης ἔοικε, φησί, τῆς ποσότητος ὁ Πλάτων τὰ πλείω ὀνόματα κατηγορῆσαι “πανδεχῆ” καὶ ἀνείδεον λέγων καὶ “ἀόρατον” καὶ “ἀπορώτατα τοῦ νοητοῦ μετειληφέναι” αὐτὴν καὶ “λογισμῷ νόθῳ μόλις ληπτήν” (15) καὶ πᾶν τὸ τούτοις ἐμφερές. αὕτη δὲ ἡ ποσότης, φησί, καὶ τοῦτο τὸ εἶδος τὸ κατὰ στέρησιν τοῦ ἑνιαίου λόγου νοούμενον τοῦ πάντας τοὺς λόγους τῶν ὄντων ἐν ἑαυτῷ περιειληφότος παραδείγματά ἐστι τῆς τῶν σωμάτων ὕλης, ἣν καὶ αὐτὴν ποσὸν καὶ τοὺς Πυθαγορείους καὶ τὸν Πλάτωνα καλεῖν ἔλεγεν, οὐ τὸ ὡς εἶδος ποσόν, ἀλλὰ τὸ κατὰ στέρησιν καὶ παρά(20)λυσιν καὶ ἔκτασιν καὶ διασπασμὸν καὶ διὰ τὴν ἀπὸ τοῦ ὄντος παράλλαξιν, δι’ ἃ καὶ κακὸν δοκεῖ ἡ ὕλη ὡς τὸ ἀγαθὸν ἀποφεύγουσα. καὶ καταλαμβάνεται ὑπ’ αὐτοῦ εἰδητικοῦ μεγέθους λόγον ἐπιδεξομένης καὶ τούτῳ ὁριζομένης, τοῦ δὲ διασπασμοῦ τῇ ἀριθμητικῇ διακρίσει εἰδοποιουμένου. ἔστιν (25) οὖν ἡ ὕλη κατὰ τοῦτον τὸν λόγον οὐδὲν ἄλλο ἢ ἡ τῶν αἰσθητῶν εἰδῶν πρὸς τὰ νοητὰ παράλλαξις παρατραπέντων ἐκεῖθεν καὶ πρὸς τὸ μὴ ὂν ὑποφερομένων (Simplicius Commentary on Aristotle’s “Physics” 9.231.5–27).
[ back ] 43. Moderatus had an affinity for Pythagorean lore, whether he agreed with it or not (Porphyry Life of Pythagoras 48). This second passage falls into three parts: a quote from Moderatus, an interjected explanation from Porphyry, and then a paraphrase of yet more of Moderatus’ account. So although we can expect the first part to reflect Moderatus’ words, the second and third parts are Porphyry’s synthesis, drawn from who knows what other parts of Moderatus’ text. Distinguishing Simplicius’, Porphyry’s, and Moderatus’ voices is admittedly difficult. For those who know the text and have puzzled over it, here is my rationale: The clearest indicators for separating the three authors are verbs of speaking. The παρατιθέμενος of lines 5–6 (Diels 1882–1895:1.231) is one of Simplicius’ favored terms for introducing extended quotations, which he tends to reproduce conscientiously. (See e.g. his quote of Geminus paraphrasing Posidonius at 291.21–292.31, which also begins with παρατίθησιν, and in which he inserts no φησι of his own.) Thus I regard line 7 onward as having no words written by Simplicius (ending, presumably, at the close of line 24, where Simplicius assesses the role of ὕλη). In that section, verbs of speaking appear at lines 7 (φησιν ὁ Πλάτων), 10 ([Πλάτων] ἐκάλεσεν), 12 ([Μοδέρατος] φησι), 13 (ὁ Πλάτων … λέγων), 15 (φησι), and 18–19 (τοὺς Πυθαγορείους καὶ τὸν Πλάτωνα καλεῖν [ὁ Μοδέρατος] ἔλεγεν). The φησι at line 15 likely has Moderatus and not Plato as its subject, parallel to the φησι at line 12. (Thus the quotation marks in my translation, distinguishing Moderatus’ voice from Porphyry’s). The change from φησι (12, 15) to ἔλεγεν (18–19) is significant, for the latter not only changes the tense but introduces indirect discourse, a sign of Porphyry’s summarizing or paraphrasing Moderatus (also in evidence in our first passage). Further, the clause in which ἔλεγεν appears (starting at line 18 with ἣν) governs the rest of the quote down through line 24, a single sentence (despite Diels’s unnecessary terminal punctuation at line 21); and this long text comprises five explanatory clauses (18: ἣν, 19: οὐ … ἀλλὰ, 20: δι’ ἃ, 22: τῆς μὲν, 24: τοῦ δὲ). So lines 18–24 contain Porphyry’s summary of Moderatus’ system, drawn from texts not necessarily near the source of the quotation in lines 7–18. For other opinions on attribution, see Dodds 1928:138n3 and Tornau 2001:204–205n26.
[ back ] 44. On the background of the concept of privation, see Tornau 2001:207–208; for its later use, Thomassen 2006:271–279. Cf. Theology of Arithmetic 9.5–6, where the dyad divides itself from the monad: πρώτη γὰρ ἡ δυὰς διεχώρισεν ἑαυτὴν ἐκ τῆς μονάδος. For scholarly opinions on the nature and function of the internal forms and λόγοι of the unitary λόγος, see Tornau 2001:209n37.
[ back ] 45. There is an important lacuna in Porphyry’s account. Note that ποσότης (feminine) is linked not with ταύτην but with τοῦτο (neuter) at line 10, a τοῦτο with no obvious antecedent, but echoed at lines 15–16: τοῦτο τὸ εἶδος (itself asking us to remember an antecedent not readily obvious). If the two τοῦτοs are to be linked, then form is a peer or immediate subordinate of quantity.
[ back ] 46. Whereas μὴ ὂν is at the bottom of Moderatus’ Platonist system, posterior or equal to the material world, in the Pythagorean system μὴ ὂν precedes it, with existence and quality as intermediaries. No doubt Moderatus and his sources were grappling with the Parmenides and the Sophist—the Platonic dialogues that deal chiefly with nonexistence. But with the loss of context, it is hazardous (and for this study tangential) to infer the reason for the difference in the two systems.
[ back ] 47. Tornau 2001.
[ back ] 48. For other relevant primary sources and discussion, see Rist 1962, Rist 1965, and Thomassen 2006:270, 275–279, and references.
[ back ] 49. Compare On Flight and Finding 94–95 (the five powers of the one God) with On Abraham 121 (two powers). The two accounts are inspired by numbers Philo encounters in Scripture; hence the disparity. Because Philo molds each account to suit the number, one should not rely upon them as maps to his metaphysics.