Cosmopoeia
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Introduction

The ancient sources

What ancient texts report about the tetractys, from Philolaus to Lucian, and how reliable this tradition is.

Not a single line by Pythagoras himself survives. Everything we know about the tetractys comes from later texts, most of them centuries younger. That does not make them worthless, but for every source one has to ask who wrote it and when.

Philolaus

The oldest Pythagorean from whom fragments of text survive is Philolaus of Croton, who lived in the fifth century BC. In one fragment (numbered 44 B 11 in the standard edition by Diels and Kranz) he praises the power of the decad: it is great and perfect, brings everything about, and is the beginning and guide of divine as well as human life. Without it, everything would remain unlimited, unclear and hidden.

The other fragments handed down under his name circle around the same ideas. In summary:

  • Limiting and unlimited. The world and everything in it is joined together from limiting and unlimited parts. If everything were unlimited, there would be nothing that could be known.
  • To know is to count. Everything knowable has number. Without number nothing can be thought or grasped.
  • Even, odd and their mixture. Number has two proper forms, the odd and the even, and a third mixed from both, the “even-odd”. How this threefold division returns in the rhythm of six of the primes is shown in The order of the primes.
  • Harmony. Because the first principles are unlike and unrelated, a harmony is needed to bind them. Philolaus describes it as a scale: the octave (1 : 2) comprises the fourth (3 : 4) and the fifth (2 : 3), and the fifth exceeds the fourth by a whole tone (8 : 9).
  • The One at the centre. The first thing fitted together, the One at the centre of the world sphere, he calls the hearth, Hestia. The One is the beginning of all things.
  • By nature, not by convention. The order of numbers is not made by humans.
  • Number does not deceive. Deception and envy belong to the unlimited and irrational. They cannot enter the nature of number and harmony.
  • Five bodies. The world sphere contains fire, water, earth and air, and as a fifth the one that encloses the sphere.

The authoritative collection of these texts is Die Fragmente der Vorsokratiker by Hermann Diels, later revised by Walther Kranz. Philolaus is listed there as number 44.

Whether the fragment on the decad really comes from Philolaus is disputed among scholars. Some consider it a later attribution. Either way it shows what rank the ten held in Pythagorean thought.

Speusippus

Speusippus, Plato’s nephew and successor at the Academy, wrote a book On Pythagorean Numbers. A longer excerpt from it survives. In it he argues why ten is the perfect number, and his reasons are geometric:

  • The numbers one to four contain point, line, surface and solid.
  • Ten contains as many primes (2, 3, 5, 7) as composite numbers (4, 6, 8, 9), if one and ten are regarded as the frame.
  • The tetrahedron, the simplest solid, can be connected with ten through its vertices, edges and faces.

Speusippus shows that already in antiquity the tetractys was understood not only as a sacred symbol but as a geometric argument.

The oath

Several later authors record an oath of the Pythagoreans. They swore not by the gods but by the one who had handed down the tetractys to them, that is, by Pythagoras. In it they call the tetractys itself the source and root of ever-flowing nature. The wording is found in the so-called Golden Verses and in Sextus Empiricus, both from long after Pythagoras.

Lucian

A vivid, if mocking, scene is told by the satirist Lucian in the second century AD. Pythagoras asks someone to count. When the man has reached four, Pythagoras interrupts: what he takes for four is in truth ten, a perfect triangle and the oath of the Pythagoreans. The joke only works if one has the tetractys in mind: counting to four already assembles the rows of the figure.

Aristotle

The most sober source is Aristotle. In his Metaphysics he reports that the Pythagoreans considered ten so perfect that they wanted the heavenly bodies to number ten as well. Since only nine could be seen (Earth, Moon, Sun, five planets and the sphere of fixed stars), they invented an invisible counter-earth. Aristotle tells this critically, as an example of how a number can overrule observation.