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] 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]