Joel Kalvesmaki, The Theology of Arithmetic: Number Symbolism in Platonism and Early Christianity
2. Generating the World of Numbers: Pythagorean and Platonist Number Symbolism in the First Century
3. The Rise of the Early Christian Theology of Arithmetic: The Valentinians
4. The Apogee of Valentinian Number Symbolism: Marcus “Magus”
5. Alternate Paths in the Early Christian Theology of Arithmetic: Monoïmus and the Paraphrase of the “Apophasis Megale”
6. The Orthodox Limits of the Theology of Arithmetic: Irenaeus of Lyons
7. The Orthodox Possibilities of the Theology of Arithmetic: Clement of Alexandria
8. How the Early Christian Theology of Arithmetic Shaped Neo-Platonism and Late Antique Christianity
Excursus A. One versus One: The Differentiation between Hen and Monad in Hellenistic and Late Antique Philosophy
Excursus B. The Pythagorean Symbol of the Τετρακτύς
Excursus C. The Dyadic Character of A Valentinian Exposition
Appendix. Greek Texts
Excursus A. One versus One: The Differentiation between Hen and Monad in Hellenistic and Late Antique Philosophy
Theon of Smyrna’s Mathematics Useful for Reading Plato, written in the second century CE, collects arithmetical, geometrical, musical, and astronomical lore relevant to Plato’s writings. In one passage, Theon summarizes various ideas about the distinction between the terms ‘one’ (ἕν) and ‘unit’ or ‘monad’ (μονάς). The passage provides important background to the ideas of the Valentinians and Clement of Alexandria, who assume that their readers are familiar with the notion of the monad’s superiority to the one. Although Theon starts off by using the terms ‘hen’ (ἕν) and ‘monad’ (μονάς) indiscriminately, he eventually turns to schools of thought that distinguished the terms.  The relative obscurity of Theon’s passage makes a full translation worth while: 
(19.7) καλεῖται δὲ μονὰς ἤτοι ἀπὸ τοῦ μένειν ἄτρεπτος καὶ μὴ ἐξίστασθαι τῆς ἑαυτῆς φύσεως· ὁσάκις γὰρ ἂν ἐφ’ ἑαυτὴν πολλα-πλασιάσωμεν τὴν μονάδα, μένει μονάς· καὶ γὰρ ἅπαξ ἓν ἕν, καὶ μέχρις ἀπείρου ἐὰν πολλαπλασιάζωμεν τὴν μονάδα, μένει μονάς. ἢ ἀπὸ τοῦ διακεκρίσθαι καὶ μεμονῶσθαι ἀπὸ τοῦ λοιποὺ πλήθους τῶν ἀριθμῶν καλεῖται μονάς.
(19.13) ᾖ δὲ διενήνοχεν ἀριθμὸς καὶ ἀριθμητόν, ταύτῃ καὶ μονὰς καὶ ἕν. ἀριθμὸς μὲν γάρ ἐστι τὸ ἐν νοητοῖς ποσόν, οἷον αὐτὰ εʹ καὶ αὐτὰ ιʹ, οὐ σώματά τινα οὐδὲ αἰσθητά, ἀλλὰ νοητά· ἀριθμητὸν δὲ τὸ ἐν αἰσθητοῖς ποσόν, ὡς ἵπποι εʹ, βόες εʹ, ἄνθρωποι εʹ. καὶ μονὰς τοίνυν ἐστὶν ἡ τοῦ ἑνὸς ἰδέα ἡ νοητή, ἥ ἐστιν ἄτομος· ἓν δὲ τὸ ἐν αἰσθητοῖς καθ’ ἑαυτὸ λεγόμενον, οἷον εἷς ἵππος, εἷς ἄνθρωπος.
(19.21) ὥστ’ εἴη ἂν ἀρχὴ τῶν μὲν ἀριθμῶν ἡ μονάς, τῶν δὲ ἀριθμητῶν τὸ ἕν· καὶ τὸ ἓν ὡς ἐν αἰσθητοῖς (20.1) τέμνεσθαί φασιν εἰς ἄπειρον, οὐχ ὡς ἀριθμὸν οὐδὲ ὡς ἀρχὴν ἀριθμοῦ, ἀλλ’ ὡς αἰσθητόν. ὥστε ἡ μὲν μονὰς νοητὴ οὖσα ἀδιαίρετος, τὸ δὲ ἓν ὡς αἰσθητὸν εἰς ἄπειρον τμητόν. καὶ τὰ ἀριθμητὰ τῶν ἀριθμῶν εἴη ἂν διαφέροντα τῷ τὰ μὲν σώματα εἶναι, τὰ δὲ ἀσώματα.
(20.5) ἁπλῶς δὲ ἀρχὰς ἀριθμῶν οἱ μὲν ὕστερόν φασι τήν τε μονάδα καὶ τὴν δυάδα, οἱ δὲ ἀπὸ Πυθαγόρου πάσας κατὰ τὸ ἑξῆς τὰς τῶν ὅρων ἐκθέσεις, δι’ ὧν ἄρτιοί τε καὶ περιττοὶ νοοῦνται, οἷον τῶν ἐν αἰσθητοῖς τριῶν ἀρχὴν τὴν τριάδα καὶ τῶν ἐν αἰσθητοῖς τεσσάρων πάντων ἀρχὴν τὴν τετράδα καὶ ἐπὶ τῶν ἄλλων ἀριθμῶν κατὰ ταὐτά.
(20.12) οἱ δὲ καὶ αὐτῶν τούτων ἀρχὴν τὴν μονάδα φασὶ καὶ τὸ ἓν πάσης ἀπηλλαγμένον διαφορᾶς ὡς ἐν ἀριθμοῖς, μόνον αὐτὸ ἕν, οὐ τὸ ἕν, τουτέστιν οὐ τόδε τὸ ποιὸν καὶ διαφοράν τινα πρὸς ἕτερον ἓν προσειληφός, ἀλλ’ αὐτὸ καθ’ αὑτὸ ἕν. οὕτω γὰρ ἂν ἀρχή τε καὶ μέτρον εἴη τῶν ὑφ’ ἑαυτὸ ὄντων, καθὸ ἕκαστον τῶν ὄντων ἓν λέγεται, μετασχὸν τῆς πρώτης τοῦ ἑνὸς οὐσίας τε καὶ ἰδέας.
(20.19) Ἀρχύτας δὲ καὶ Φιλόλαος ἀδιαφόρως τὸ ἓν καὶ μονάδα καλοῦσι καὶ τὴν μονάδα ἕν.
(20.20) οἱ δὲ πλεῖστοι προστιθέασι τῷ μονάδα αὐτὴν τὴν πρώτην μονάδα, ὡς οὔσης τινὸς οὐ πρώτης μονάδος, ἥ ἐστι κοινότερον καὶ αὐτὴ μονὰς καὶ ἕν—λέγουσι δὴ καὶ τὸ ἕν  —τουτ(21.1)έστιν ἡ πρώτη καὶ νοητὴ οὐσία τοῦ ἑνός, ἑκάστου τῶν πραγμάτων παρέχουσα ἕν· μετοχῇ γὰρ αὐτῆς ἕκαστον ἓν καλεῖται. διὸ καὶ τοὔνομα αὐτοῦ οὐδὲν παρεμφαίνει τί ἓν καὶ τίνος γένους, κατὰ πάντων δὲ κατηγορεῖται, [ὥστε καὶ ἡ μονὰς καὶ ἕν ἐστι,] κἂν τὰ μὲν νοητὰ καὶ παραδείγματα μηδὲν ἀλλήλων διαφέροντα, τὰ δὲ αἰσθητά.
(21.7) ἔνιοι δὲ ἑτέραν διαφορὰν τῆς μονάδος καὶ τοῦ ἑνὸς παρέδοσαν. τὸ μὲν γὰρ ἓν οὔτε κατ’ οὐσίαν ἀλλοιοῦται, οὔτε τῇ μονάδι καὶ τοῖς περιττοῖς αἴτιόν ἐστι τοῦ μὴ ἀλλοιοῦσθαι κατ’ οὐσίαν, οὔτε κατὰ ποιότητα, αὐτὸ γὰρ μονάς ἐστι καὶ οὐχ ὥσπερ αἱ μονάδες πολλαί, οὔτε κατὰ τὸ ποσόν· οὐδὲ γὰρ συντίθεται ὥσπερ αἱ μονάδες ἄλλῃ μονάδι· ἓν γάρ ἐστι καὶ οὐ πολλά, διὸ καὶ ἑνικῶς καλεῖται ἕν.
(21.14) καὶ γὰρ εἰ παρὰ Πλάτωνι ἑνάδες εἴρηνται ἐν Φιλήβῳ, οὐ παρὰ τὸ ἓν ἐλέχθησαν, ἀλλὰ παρὰ τὴν ἑνάδα, ἥτις ἐστὶ μονὰς μετοχῇ τοῦ ἑνός. κατὰ πάντα δὴ ἀμετάβλητον τὸ ἓν τὸ ὡρισμένον τοῦτο ἐν τῇ μονάδι. ὥστε διαφέροι ἂν τὸ ἓν τῆς μονάδος, ὅτι τὸ μέν ἐστιν ὡρισμένον καὶ πέρας, αἱ δὲ μονάδες ἄπειροι καὶ ἀόριστοι.
(19.7) It is called ‘monad’ either from remaining unchangeable and not departing from its own nature.  For however often we multiply the monad against itself, it remains monad. For one once is one, and should we multiply the monad ad infinitum, it remains a monad. Or it is called ‘monad’ from being distinguished and isolated from the rest of the multitude of numbers. 
(19.13) As number differs from numerable thing, so monad differs from one. For number is intelligible quantity, for example, five itself and ten itself, not certain bodies or sense-perceptible objects, but intelligible objects. But a numerable thing is sense-perceptible quantity, for example, five horses, five cows, five people. And so a monad, which is indivisible, is the intelligible form of the one. A sense-perceptible one is spoken of absolutely, for example, “one horse,” “one person.”
(19.21) Thus, the monad would be the origin of numbers, but the one, of numerable things. And they say that the sense-perceptible one (20) is divided ad infinitum,  neither qua number nor qua origin of number but qua sense perceptible. So the monad, being intelligible, is indivisible but the one, as a sense perceptible, is infinitely divisible. And numerable things would differ from numbers in that the former are embodied, and the latter without bodies.
(20.5) Some more recently say simply that the monad and dyad are the origins of numbers, whereas the Pythagoreans [assign this to] the entire subsequent series of limits, through which even and odd are conceived of;  for example, the triad is the origin of [all] sense-perceptible threes and the tetrad is the source of every sense-perceptible four, and likewise for the other numbers.
(20.12) Others say that the origin of these very things is the monad and the one removed from every difference that occurs in numbers—only one itself, not the one, that is, not the one exhibiting this quality and certain difference toward another one, but absolute one. So it would be the origin and measure of entities [generated] by itself, by which each entity is called “one,” participating in the one’s primary substance and form.
(20.19) Archytas and Philolaus call the one ‘monad’ and the monad ‘one,’ without differentiation.
(20.20) The majority include the primary monad with monad itself, since there is a certain monad that is not primary, but is more commonplace and is monad itself and one—and indeed, they call it the one, that is, (21) the primary and intelligible substance of the one, furnishing [the attribute] one to each thing. For each thing is called ‘one’ by participation in it. Wherefore its name suggests nothing about what is “one” and of what sort, but it is predicated of everything , whether they be intelligible object and paradigms (which do not differ from each other), or sense-perceptible objects.
(21.7) Some hand down a different distinction between the monad and the one. For the one neither changes in substance (nor is it the cause of the monad’s and odd numbers’ being <un?>alterable in substance), nor [does it change] in quality (for it is a monad, and is not like many monads), nor [does it change] in quantity (for it is not added to another monad, like monads [are]). For it is one and not many, wherefore it is called ‘one’ in a unifying manner.
(21.14) For even if henads have been mentioned by Plato in Philebus, they weren’t said [to be] in distinction to the one, but rather in distinction to the henad, which is a monad by virtue of participation in the one. Indeed, in respect to everything in the monad this defined one is unchangeable. So the one would differ from the monad in that the former is defined and limited, whereas monads are infinite and undefined.
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.’  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).  Numerables, on the other hand, are “quantity in sense-perceptibles,” and are predicated of physical objects (19.17).  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.  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.’  There is no evidence that Plato distinguished the terms, either.  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.  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).  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.  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.  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.  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.  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. 
Philo uses the terminological distinction to make a theological point about Genesis 24.22.  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.  Clement of Alexandria, seemingly inspired by Philo, embraces the hierarchy hen [θεός] → monad → hen. 
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).  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).  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. 
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.
[ back ] 1. 18.5 vs. 18.11, 14; 19.6 vs. 19.7.
[ back ] 2. Text in angle brackets is excised by the modern editor. Words in square brackets are insertions, by the editor or me, for sense.
[ back ] 3. Hiller 1878:20: καὶ τὸ ἕν] οὐ τὸ ἕν?
[ back ] 4. This sentence is paralleled in Stobaeus Eclogae 1.1.8, attributed to Moderatus of Gades (fl. first c. CE). Underlines here highlight his etymology.
[ back ] 5. This sentence is also paralleled in Stobaeus.
[ back ] 6. The text from the beginning of the paragraph to this point is paralleled in Stobaeus.
[ back ] 7. From the previous sentence, “And numerable things … ,” to this point is paralleled in Stobaeus.
[ back ] 8. Theon has been tacitly following Moderatus. Two of the three fragments of Moderatus preserved by Stobaeus have parallels in Theon. Moderatus fragment 1 = Theon 18.3–9 + 19.7–8 + 19.12–13. Moderatus fragment 2 = Theon 19.21–20.1 + 20.4–9. Dodds argues that Theon 19.15 depends on Moderatus (1928:138). Full analysis—indeed a complete corpus of Moderatus’ literary fragments—is still needed.
[ back ] 9. ἀριθμὸς μὲν γάρ ἐστι τὸ ἐν νοητοῖς ποσόν. See also 21.5 and the fragment of Moderatus cited by Simplicius, discussed in chapter 2 above. The distinction between number and numerable thing is maintained by Porphyry, who compares it to the distinction between harmony and something harmonized (Commentary on Ptolemy’s “Harmonics” 12.2–5).
[ back ] 10. ἀριθμητὸν δὲ τὸ ἐν αἰσθητοῖς ποσόν. See also 21.6.
[ back ] 11. See chap. 2 above.
[ back ] 12. Syrianus Commentary on Aristotle’s “Metaphysics” 151.17–22; Aristotle, fragment 203, in Alexander of Aphrodisias Commentary on the “Metaphysics” 39.15.
[ back ] 13. Note, for instance, the preponderance of ‘hen’ in the Parmenides, and comparative lack of interest in ‘monad’ as a technical term. The late antique writers who read Plato’s writings most closely rely nearly exclusively on ‘hen’ to describe the metaphysics of arithmetic. Plotinus, for instance, nearly always uses ‘hen,’ not ‘monad,’ to describe all his various metaphysical levels of number, in conformity with Plato’s Parmenides. See Edwards 2006:65–72.
[ back ] 14. See the critical apparatus in Hiller 1878:20–21 for the serious textual problems.
[ back ] 15. This follows the suggested emendation of Ismaël Bullialdus in the critical apparatus of Hiller’s ed. Without this emendation, the parallelism of contrasts in lines 9–14 is broken.
[ back ] 16. Aristotle Metaphysics A6, 987b14–18.
[ back ] 17. According to my reading of 21.8–13, the punctuation in Hiller’s edition should be emended, converting the first comma in line 10 and the comma in line 11 to colons (·) and the colon in line 12 to a comma.
[ back ] 18. As well as the reports listed here, see Sextus Empiricus Against the Physicians 2.261; pseudo-Pythagoras in pseudo-Justin Martyr (III) Exhortation to the Nations 19.2 (ed. Otto 1879:18c); Favonius Eulogius Disputation on the Dream of Scipio 3.1–31; John Lydus On the Months 2.6; Proclus Commentary on the “Timaeus” 1:16.27–29; Boethius De unitate et de uno; Asclepius of Tralles Commentary on Nicomachus of Gerasa 41.
[ back ] 19. Alexander Polyhistor, fragment 140 (ed. Müller 1849:240b), in Diogenes Laertius Lives of the Philosophers 8.24–25. See also Dillon 1996:127. Alexander agrees with an undatable Pythagorean text ascribed to Xenocrates, who uses ‘first monad’ in place of ‘hen’ (“Xenocrates,” fragment 120.77, in Sextus Empiricus Against the Physicians 2.261–262). For other late antique uses of ‘monad’ instead of ‘hen,’ see also idem, 2.282 and Aetius Placita 281.5.
[ back ] 20. Sextus Empiricus Against the Physicians 2.276.
[ back ] 21. Philo Questions and Answers on Genesis 4.110.
[ back ] 22. Philo, at Who Is the Heir of Divine Things? 187–190, describes the monad as source of numbers, but does not contrast it to the hen. In On Rewards and Punishments 41, he uses ‘hen’ and ‘monad’ as a pair, but it is unclear whether he is distinguishing or conflating the terms (cf. idem On the Unchangeableness of God 11). At On the Creation of the World 98 he uses ‘hen’ where ‘monad’ might be expected; at On Abraham 122 he uses ‘monad’ where ‘hen’ might be called for. At Allegorical Interpretation 2.3 he uses both terms together, but specifies that the “one God” (hen theon) supersedes the monad. This may be Philo’s way of using the language of “one God,” native to Judaism, to invert and thereby challenge the monad → hen doctrine so directly stated at Questions and Answers on Genesis 4.110. See also p. 127 above.
[ back ] 23. Clement Paedagogue 1.8.71, discussed at p. 126 above.
[ back ] 24. Refutation of All Heresies 1.2.9 = 4.51.7.
[ back ] 25. Refutation of All Heresies 1.2.6 = 4.51.4.
[ back ] 26. See Refutation of All Heresies 1.2.2, 6.23.1.