The Theology of Arithmetic: Number Symbolism in Platonism and Early Christianity

  Kalvesmaki, Joel. 2013. The Theology of Arithmetic: Number Symbolism in Platonism and Early Christianity. Hellenic Studies Series 59. Washington, DC: Center for Hellenic Studies.

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. [1] The relative obscurity of Theon’s passage makes a full translation worth while: [2]

(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.

(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.

(19.13) 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.”

(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 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.

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.’

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.