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.
Excursus B. The Pythagorean Symbol of the Τετρακτύς
The Pythagoreans symbolized the number ten with a special term for the first four numbers, the τετρακτύς. [1] The term is probably of Doric origin, but it is unclear exactly how this unusual word, which comes from the root meaning “four,” was coined. [2] The term is first attested in texts from the first century, [3] quoting earlier, but not precisely datable, Pythagorean texts. Some may go back to the shadowy origins of early Pythagoreanism, others may be precursors to its reinvention in the late Roman republic. The concept underlying the τετρακτύς has been shown to have predated Pythagoras in non-Greek societies. [4]
The τετρακτύς refers to the first four numbers, which were depicted in the Pythagorean tradition as four rows of ten pebbles arranged in the shape of an isosceles triangle: 
The figure symbolizes a collection in unity. It also emphasizes that the sum of the first four numbers is ten, a number revered as constituting the foundation of all numbers. By depicting a harmonious arrangement of pebbles, the figure demonstrates the complementary character of arithmetic and geometry, one of the trademarks of the quadrivium. This triangular figure was so well known in the Hellenistic period that Lucian, in one of his satires on the philosophers, has Pythagoras instruct a prospective “buyer” of philosophy to count to four. This four, says Pythagoras, “is ten, and a perfect triangle, and our oath.” [5] The oath in question is found in the so-called Golden Poem, attributed to Pythagoras and probably the oldest of the Pythagorean texts to mention the τετρακτύς: [6]
The figure symbolizes a collection in unity. It also emphasizes that the sum of the first four numbers is ten, a number revered as constituting the foundation of all numbers. By depicting a harmonious arrangement of pebbles, the figure demonstrates the complementary character of arithmetic and geometry, one of the trademarks of the quadrivium. This triangular figure was so well known in the Hellenistic period that Lucian, in one of his satires on the philosophers, has Pythagoras instruct a prospective “buyer” of philosophy to count to four. This four, says Pythagoras, “is ten, and a perfect triangle, and our oath.” [5] The oath in question is found in the so-called Golden Poem, attributed to Pythagoras and probably the oldest of the Pythagorean texts to mention the τετρακτύς: [6]
οὐ μὰ τὸν ἁμετέρᾳ κεφαλᾷ παραδόντα τετρακτύν,
παγὰν ἀενάου φύσεως ῥιζώματ᾽ ἔχουσαν
No, by the one who grants our head the τετρακτύς,
Fount possessing roots of everlasting nature. [7]
παγὰν ἀενάου φύσεως ῥιζώματ᾽ ἔχουσαν
No, by the one who grants our head the τετρακτύς,
Fount possessing roots of everlasting nature. [7]
In light of the authentic fragments of Philolaus, these two lines can be reasonably interpreted to suggest that the ancient Pythagoreans held that the first four numbers were forged out of the principles of nature (in Philolaus, these are ‘limiters’ and ‘unlimiteds’) to provide a “spring” for the physical world. There is the intriguing possibility that the couplet comes from the same literary milieu as Philolaus’ lost work On Nature. [8]
In the first and the second centuries, probably as a result of the revival and reinvention of Pythagoreanism, the τετρακτύς entered non-Pythagorean literary circles as a metaphor. [9] Because legend had it that the Pythagorean tradition was a secret one, and because the τετρακτύς was seen as the basis of their oath, the symbol took on special mystical significance that extended beyond its primary mathematical meaning. Like other Pythagorean symbols, it could connect dis-parate foursomes in the world. Theon of Smyrna collected eleven different quaternities found in the world, calling them τετρακτύες. [10] His examples range from the mathematical (point, line, plane, and solid) to anthropological (ages of human beings: child, teenager, adult, and elder). Another author, of unknown date, drew up a similar list of six τετρακτύες, three of which have no parallels in Theon or other ancient authors. [11] This reflects the popular, literary character of Pythagoreanism. An author could theoretically take any foursome, relate it to the τετρακτύς, and thereby tap into the world of Pythagorean symbolism.
Similarly, an author could postulate a foursome and describe its internal relations so as to invoke Pythagorean overtones. In return, Pythagorean imagery and terminology used to structure a sequence of four elements could reinforce and supplement the lore of the τετρακτύς. In philosophy and theology in late antiquity this phenomenon is common. The internal structure of a philosophical or theological quaternity generally follows one of two patterns. In the first, the initial element of the quaternity begets the second, the second begets the third, and the third the fourth—a linear progression. Examples of this kind of quaternity are the number series one, two, three, and four; or the geometric series point, line, plane, and solid. [12] In the second pattern, the author conceives of the foursome as two complementary, hierarchical pairs. The pairs can be expressed in the relation A : B :: C : D, like the four corners of a rectangle. Examples of this are Neoplatonic theories of epistemology or the quadrivium.
Polemic aside, the accusations of Irenaeus, Hippolytus, and other apologists who charge the Valentinians and others of teaching the Pythagorean τετρακτύς in the guise of Christian doctrine cannot be dismissed as fraudulent. The apologists’ rhetoric is often excessive, as expected in the genre. But they correctly recognize that certain doctrines—for example the Monotes-Henotes-Monas-Hen doctrine assigned to an unnamed Valentinian and Marcus—are attuned to a Pythagorean model of foursomes. [13]
The use of τετρακτύς in Christian literature reflects the early but transient suspicion the orthodox had of gnostic opponents. In the second and third century, orthodox Christian authors use τετρακτύς in a disparaging or neutral way. [14] But in the fourth century, after Valentinianism waned, Christians freely used it to symbolize Christian truths, such as the unity of the four Gospels and the fourfold character of Christian virtue. [15]
Footnotes
[ back ] 1. A number of studies have been published on the τετρακτύς. The most extensive are Delatte 1915:249–268 and Kucharski 1952. See also Apatow 1999, Sbordone 1981, Lampropoulou 1975, Burkert 1972 passim, and Haase 1969. On the special use of the term in music see Kárpáti 1993.
[ back ] 2. See Burkert 1972:222n24 and Delatte 1915:253–254. Cf. Chalcidius Commentary on the “Timaeus” 35 (84.9–11), who calls it the quadratura.
[ back ] 3. See n9 below.
[ back ] 4. See Burkert 1972:474n50.
[ back ] 5. Lucian Vitarum auctio 4: ἃ σὺ δοκέεις τέσσαρα, ταῦτα δέκα ἐστὶ καὶ τρίγωνον ἐντελὲς καὶ ἡμέτερον ὅρκιον. Other explanations of the τετρακτύς as the summation of the first four numbers are found in Aetius Placita 1.3.8 (= Diels and Kranz 1957:58b.15); Sextus Empiricus Against the Logicians (= Against the Mathematicians 6–7) 1.94; and Hippolytus Refutation of All Heresies 1.2.8, 4.51.6, 6.23.2–5.
[ back ] 6. On the poem consult Derron 1992. Delatte traces this fragment to Timaeus, of the fourth c. BCE, and an anonymous treatise on arithmology of the second or third c. BCE (1915:249–253). The τετρακτύς is also attested in the ακούσματα of the Pythagoreans in Iamblichus The Pythagorean Way of Life 82.12 (Diels and Kranz 1957:58c.4). Earlier, Hellenistic Pythagorean texts that mention the τετρακτύς are “Lysis,” fragment 4.4 (= Diels and Kranz 1957:46.4), in Athenagoras Legatio 6.1, and an anonymous philosopher paraphrased in Photius Bibliotheca 439a7–8 (Bekker 1825). Thesleff 1961 tentatively dates these to the fourth and third c. BCE, respectively. Also see Philolaus, fragment 11 (found in Lucian De lapsu in salutandum 5), of dubious date and authenticity.
[ back ] 7. Sextus Empiricus Against the Logicians (= Against the Mathematicians 6–7) 1.94. The two lines are reproduced with significant differences in other authors: pseudo-Pythagoras Golden Poem 47–48; Aetius Placita 282.3–7; Nicomachus of Gerasa, in Theology of Arithmetic 22.21–22; Sextus Empiricus Against the Mathematicians 4.2; Theon of Smyrna Mathematics Useful for Reading Plato 94.6–7; Hippolytus Refutation of All Heresies 6.23.4; Porphyry Life of Pythagoras 20.18–19; Iamblichus The Pythagorean Way of Life 29.162.17–18; Julian To the Untaught Dogs 15.34; Stobaeus Eclogae 1.10.12.72–73; Hierocles On the “Golden Poem” 20; Damascius Commentary on the “Parmenides” 63.29; Proclus Commentary on the “Timaeus” 2.53.6. For analysis of these differences, see Delatte 1915:249–253. Possibly even Xenocrates (frags. 101–102, Isnarde Parente 1982), when he suggests that “the universe consists of the One and the Everlasting” (συνεστάναι τὸ πᾶν ἐκ τοῦ ἑνὸς καὶ τοῦ ἀενάου), uses the Pythagorean τετρακτύς as a symbol of matter. Such an ancient testimony does not help date the Golden Poem, but it does establish the antiquity of the motif. See Dillon 1996:24.
[ back ] 8. The argument in outline is this: Philolaus is concerned with “nature,” an important concept in the couplet. One may interpret the second line to say that the τετρακτύς is the root of eternal nature. But it is equally possible to read the genitives so that the roots producing the τετρακτύς derive from eternal nature. In this case, number is subordinate to and derived from eternal principles such as unlimiteds and limiters, which as Huffman has stressed is the nature of Philolaus’ philosophy (1993). Further, the epithets for the τετρακτύς in the ακούσματα—the oracle at Delphi, harmony, and the location of the Sirens (Iamblichus The Pythagorean Way of Life 82.12 [= DK 58c.4])—are ancient, nonmathematical, and (with the exception of ‘harmony’) nonphilosophical. Does the couplet, then, derive from the μαθηματικοί faction of ancient Pythagoreanism (see Burkert 1972)? If so, the question of origins remains: did Philolaus write the couplet, was the couplet composed in light of Philolaus’ book, did the couplet come from the older Pythagorean oral tradition, or are the couplet and Philolaus independent of each other but dependent upon a common tradition?
[ back ] 9. Search results from the TLG (E) are instructive on the popularity of the term. Discounting the statistics for the Hellenistic Pythagorean texts (very difficult to date), τετρακτύς appears in no texts BCE, five times in the first century CE, forty-six in the second, twenty-six in the third, twenty-four in the fourth, and twenty-four in the fifth.
[ back ] 10. Mathematics Useful for Reading Plato 94–99.
[ back ] 11. This very brief treatise, called Τετρακτὺν τὴν τὰ πάντα διατείνουσαν καὶ διοιροῦσαν τετραχῆ πάντα (The Tetraktys Suspending and Apportioning All Things Four-fold), is found in Paris gr. 1185 suppl., fol. 62v., and is published in Delatte 1915:187.
[ back ] 12. Cf. the examples provided by Alexander Polyhistor and Hippolytus, discussed at pp. 181–182 above.
[ back ] 13. See pp. 31–33 and 75 above.
[ back ] 14. The most favorable religious use is by Athenagoras Legatio 6.1, who simply presents it as a part of the philosophical apparatus that undermines polytheism. Other instances are found at Irenaeus Against Heresies 1.1.1, 1.1.13, 1.8.4–5, 1.8.10, 1.11.1–2; Clement of Alexandria Stromateis 2.23.138.6; Hippolytus Refutation of All Heresies 1.2.9; 4.51.7; 6.23.4–5; 6.24.1; 6.34.1; 6.44.1; 6.45.2; and Anatolius of Laodicea On the Decad 5.11, 8.1, 15.20.
[ back ] 15. Eusebius of Caesarea Church History 3.25.1; Theodoret of Cyrus Letters 131.112, 146.200; Evagrius of Pontus On Prayer pref. (PG 79.1165d).