Excursus A. One versus One: The Differentiation between Hen and Monad in Hellenistic and Late Antique Philosophy

Thus Theon, almost certainly following Moderatus, at least in the beginning, explains the etymology of ‘monad,’ and then offers six opinions concerning the difference between ‘monad’ and ‘hen.’ [8] At 19.13, the first of these explanations, he relates the terms to the difference between number (ἀριθμός) and numerable thing (ἀριθμητόν). In distinguishing number from numerables, Theon defines the former as intelligible quantity (literally “quantity in intelligibles”), and not part of the material world (19.15). [9] Numerables, on the other hand, are “quantity in sense-perceptibles,” and are predicated of physical objects (19.17). [10] Numerables have bodies, but numbers are bodiless (19.16, 20.5). As numbers are to numerables, Theon/Moderatus claims, so is the monad to the hen (19.13–15). The monad is the intelligible form of the hen, and is indivisible (19.19). Both the monad and the hen are principles: the monad of numbers, and the hen of numerables (19.21–22). The monad and the hen differ, too, in that only the hen may be divided infinitely, because it resides in the corporeal world (19.22–20.4).

Thus we have in this system of thought the notion that the monad stands metaphysically over the hen, with each of the two presiding as the first principle of everything else on its level. The monad presides over objects of intellection; the hen over sense perception.

Theon, again paralleling Moderatus, claims that the next group, more “recent” than the first, identified simply the monad and dyad as the principles of numbers, unlike the Pythagoreans, who claimed that all the intellectual numbers—monad, dyad, triad, tetrad, and so on—provided the principles for the numbers instantiated in the realm of sense perception—hen, duo, tria, tettares, and so on (20.5–11). The contrast has echoes in Eudorus, who also contrasts monistic with allegedly later dualist Pythagoreans. [11] But the texts upon which this comparison is based are too short and vague to make the association definite.

A third, unnamed group, according to Theon, claims that the monad and the hen—not just the hen as a quality or point of differentiation, but the absolute hen—were the principle and measure of beings (20.12–19). This absolute one or monad lends its primary substance and form to entities, whereby they can be said to be one. Thus in this monadic system the contrasting terms are ‘monad’/‘absolute hen’ and ‘hen.’

Theon presents yet a fourth group, consisting of Archytas and Philolaus, who he says make no distinction between hen and monad (20.19–20 = Archytas, test. 20 = Philolaus, test. 10). Although Syrianus (fl. 430s CE) contradicts him, Theon is probably correct, since Aristotle, one of the more reliable sources for pre-Platonic Pythagoreanism, states that the Pythagoreans called νοῦς both ‘monad’ and ‘hen.’ [12] There is no evidence that Plato distinguished the terms, either. [13] Theon, therefore, confirms that the distinction became current only after Plato, and had no currency in classical Pythagoreanism, and that this was still known in his time.

Theon presents a fifth group—the “majority”—that probably overlaps with some of the previous groups. The text is rather muddled, but enough is clear to know that they distinguished one kind of monad from another. [14] They call the lower monad “more common,” “monad itself,” and “hen” but not “the hen.” This is reserved for the higher monad, the chief intelligible essence that furnishes to individual things the property of being one (20.20–21.2). For this group, like the third, something can be said to be “one” by virtue of its participation in the monad (21.2–3). The term ‘one’ is merely a predicate, an indication of something numerable, whether it be in the intellectual or the sense-perceptible realm. Thus, like the first group, they embrace the hierarchy monad → hen, where the arrow indicates not only metaphysical priority, but a transfer of properties. The system also suggests that the first, absolute monad presides over a realm of intellectual paradigms, which itself presides over the realm of sense per-ception.

A sixth group distinguishes between ‘hen’ and ‘monad’ in a different way. To them, the superiority of the hen to the monads (note the plural) is manifest in its threefold immutability. The hen is immutable in its essence, an immutability that cannot be attributed to the monad or to the odd numbers (21.7–10). [15] Second, the hen is immutable in its quality since it is a monad and is unlike many monads (21.10–11). This vague phrase may refer to the Platonic distinction between ideal numbers, which are unique, and intermediate mathematicals, which resemble each other. [16] Third, the hen is immutable in quantity since it cannot be added the way one monad is combined with another. Otherwise the hen would be many and no longer one (21.11–13). So the monads, treated as countables, change in numerical identity as they mathematically combine. Over the monads resides the hen (occasionally described as a monad). The three immutable aspects of this hen—essence, quality, and quantity—correspond to the first three of Aristotle’s categories, pointing to a school of philosophy with strong interest in both Plato and Aristotle. [17] For this group, the ultimate distinction between hen and monad is that the former is defined and is a limit, whereas monads are limitless and indefinite. This arrangement, hen → monad, reverses the schemes found in other groups Theon discusses.

Theon’s survey concisely presents the various distinctions made in the second century between ‘hen’ and ‘monad,’ and the importance assigned to the subject. To Theon’s doxography can be added several other systems from roughly the same period. [18] Alexander Polyhistor, who recounts the Pythagorean doctrine of the generation of numbers, describes the monad’s begetting the indefinite dyad, which in turn generates other numbers. Here the monad altogether supplants the traditional hen. [19] Sextus Empiricus too quotes a Pythago-rean source that holds to the “first monad” and the “indefinite dyad” as the first principles. The hen derives from this first monad, whereas the number two emerges from the combination of the monad and the indefinite dyad. [20]

Philo uses the terminological distinction to make a theological point about Genesis 24.22. [21] He notes that the monad is to the hen as the archetype is to the copy, and he does so presupposing that this analogy is common knowledge. Although he attests to the doctrine’s wide distribution, Philo does not consistently hold to the scheme monad → hen. [22] Clement of Alexandria, seemingly inspired by Philo, embraces the hierarchy hen [θεός] → monad → hen. [23]

Hippolytus reports a rather strange version of the hen → monad doctrine when he claims that the Pythagoreans held to the hierarchy number → monad → n² → n³ (ἀριθμός, μονάς, δύναμις, κύβος). [24] In this arrangement, the hen is associated with the level of “number,” and the first monad is the principle of numbers “in their instantiation” (καθ ᾽ ὑπόστασιν). [25] All four levels are associa-ted with the τετρακτύς, and are thought of as the four parts of the decad. This series also corresponds to that of point, line, plane, and solid. But Hippolytus does not consistently rely upon this scheme. Elsewhere, he slips into language that prioritizes the monad. [26]

Overall, then, late antique authors frequently distinguished between the terms ‘monad’ and ‘hen,’ and they took quite different approaches. Although many considered the monad superior to the hen, the variety of opinion shows that there was no consensus, merely a lively interest. Some opinions were an intricate part of an author’s overall philosophical commitment. Especially notable among those who most emphasized the distinction was a belief in multiple levels of immaterial reality, particularly levels of mathematicals. The distinction between ‘hen’ and ‘monad’ helped to articulate that hierarchy.