Perhaps I’m simply becoming provocative for the sake of provoking; or maybe I’m losing my mind; but in the process of writing this book on mathematical Pythagoreanism and its supposed progenitor Hippasus of Metapontum (fl. early-mid 5th Century BCE), I am beginning to see mathematical Pythagoreanism everywhere. And I mean everywhere (even the New York Times).
Witness Democritus of Abdera, father of atomism, and – dare I say it! – mathematical Pythagorean.
It all started when I was researching pre-Aristotelian ‘predicative’ uses of the verb ὑπαρχεῖν for a talk I will be giving at the University of Toronto in April, when I discovered that this verb does not occur before mathematical Pythagorean Philolaus of Croton in philosophical contexts – not even among the Eleatics – except for two places: in a fascinating and bizarre fragment of the Sicilian comedian Epicharmus (which expresses mathematics in a philosophical way), and in Democritus’ longest surviving fragment (DK 68 B 191), as preserved by Stobaeus. Now the use of ὑπαρχεῖν by Democritus is not as exciting as that of Epicharmus – if you want to hear about that, come see my talk at the Collaborative Programme in Ancient and Medieval Philosophy at the University of Toronto on April 8.
But something else set off my alarm about the Democritus fragment – its potential as a source for our understanding of the ‘Pythagorean’ theory of the soul as a material harmony, as described famously by Simmias in Plato’s Phaedo (86b). Here’s Democritus:
For men achieve cheerfulness by measuredness in pleasure and by proportioning of life; excess and deficiency are apt to fluctuate and cause great movements in the soul. And motions of the souls away from great intervals are neither stable nor cheerful.
ἀνθρώποισι γὰρ εὐθυμίη γίνεται μετριότητι τέρψιος καὶ βίου συμμμετρίῃ· τὰ δ’ἐλλείποντα καὶ ὑπερβάλλοντα μεταπίπτειν τε φιλεῖ καὶ μεγάλας κινήσιας ἐμποιεῖν τῇ ψυχῇ. αἱ δ’ ἐκ μεγάλων διαστημάτων κινούμεναι τῶν ψυχέων οὔτε εὐσταθέες εἰσιν οὔτε εὔθυμοι.
(Democritus, D 55 Taylor, DK 68 B 191 = Stobaeus, Eclogae III.1.210)
It’s pretty obvious that Democritus is describing a system of moral psychology in terms related both to mathematics and to natural science: cheerfulness, which is the aim of one’s life, “comes to be” by means of “measuredness in pleasure” and “proportioning of life”; concurrently, engaging in behavior marked by “excess and deficiency” – apparently away from the ideal of “measuredness” – provokes “great movements” in the soul. What are these “movements”, we might ask? Democritus goes on: they are the souls’ “motions away from great intervals”, which are said to hinder stability and cheerfulness. What the frap does that mean?
Taylor (1999, p.25) doesn’t offer much help: he renders the passage “souls which change over great intervals”, following Vlastos. Vlastos (1995, p. 334), for his part, glosses by claiming that the “opposite to the ‘great movements’…would therefore be a dynamic equilibrium’ and compares this passage with others in the Hippocratic treatises, which is surely right, but doesn’t explain the whole “intervals” part. Brancacci and Morel’s new collected volume, called Democritus: Science, the Arts, and the Care of the Soul, doesn’t even mention this bit. And yet there must be something significant here about that appeal to “measurement”, “proportioning”, “excess and deficiency”…
I go in search of Democritus’ mathematical knowledge and psychology by appeal to the lowest common denominator in the history of philosophy, Diogenes Laertius, and imagine my surprise when this is what I see:
It is also clear from his writings what sort of person [Democritus] was. He seems, says Thrasyllus, to have been an adherent of the Pythagorean doctrines; and indeed he refers to Pythagoras himself, expressing admiration of him in his book of the same title. He appears to take all his views from him, and would even appear to have been his pupil, did not chronology make that impossible. But Glaucus of Rhegium, who lived about the same time, says that [Democritus] had all his teaching from one of the Pythagoreans, and Apollodorus of Cyzicus says he was an associate of Philolaus.
(Diogenes Laertius IX.38, Trans. by Taylor)
At this point, everything begins to fall into place. Glaucus of Rhegium was a performing musician and one of the very first historians of music who was active around the end of the 5th Century BCE (roughly same time as Democritus and Philolaus, and when Archytas and Plato were young); and among other things, he was said by Aristoxenus (student of Aristotle, late 4th Century BCE, as reliable an authority on the early history of music as there is; F 90 Wehrli) to have taken up Hippasus of Metapontum’s project of playing around with bronze discs (on Glaucus, see Barker 2007: 84-86). According to Aristoxenus, that project involved striking four discs of the same diameter but differing in thickness, such that the ratio of thickness between the first and the other three discs (respectively) was 4:3, 3:2, and 2:1. What happened? When Hippasus struck them together, they produced the pitches that occur at a perfect fourth, a perfect fifth, and an octave. Musical pitch was now understood as a matter of ratios between integers, and the old idea that there might be a harmonic structure to the universe – an idea as old as Heraclitus, and perhaps espoused by Pythagoras too – that could be rendered in arithmetical terms became tenable.
What did Hippasus call these ratios of integers? We don’t know, since nothing from his writings (if indeed he wrote down anything) survives; but both Philolaus and Archytas after him took up the project of trying to make sense of harmonics, and each made significant contributions to that project too. For his part, Archytas (F 2 Huffman, quoted directly by Porphyry in On Ptolemy’s Harmonics 1.5) called those ratios “intervals” (διαστημάτα) and sought to understand how the speed of things in motion, e.g. rhomboi, correlated with pitch; he also encouraged ethical moderation (T A9 and A9a Huffman); and the concept of “calmness” (γαλήνη) played a significant role in his philosophy (T A22), as it might have in Democritus’ philosophy as well (DK 68 A1 = D.L. IX.45).
Glaucus of Rhegium was one of the very first compilers of musical history in Greece, and he knew both about Hippasus’ discs and about Democritus’ involvement with Pythagoreans. We know that Democritus undertook investigations into music theory himself: he wrote four books on this, of which the first, called On Rhythms and Harmony, had an historical component that was heurematographical (cf. Brancacci 2007, pp. 186-88). Both rhythm and harmony had important roles to play in Democritus’ general atomistic physics as well as his psychology (see, e.g., DK 68 A167).
We are thus stimulated to take very seriously the idea that, when Democritus refers to “motions of the souls away from great intervals”, he is referring to the musical “intervals” that a soul possesses when at rest and cheerful. Motions that deviate from the “great intervals” are either excessive or deficient and cause psychological consternation.
Two important consequences follow from this reinterpretation of the longest surviving fragment of Democritus: first, that he cannot be counted out as a reference when Simmias refers to a theory of the soul as a material harmony in the Phaedo, on the grounds that he espoused a material psychology that was at very least a modification of something Pythagorean, and most likely Philolaic; and second, that when Glaucus of Rhegium spoke of Democritus as deriving his learning from “one of the Pythagoreans”, a candidate for this teacher would be Hippasus of Metapontum, who was roughly one generation earlier than Democritus (although there is the testimony of Duris of Samos [= DK 14 A6, not in Taylor] that refers to Arimnestos as the teacher of Democritus). If I am right, then we have reason to entertain the possibility that Democritus may have had strong associations with the mathematical Pythagorean group whose philosophy concerned itself with the mathematics of nature.
Would this make Democritus a Pythagorean?
Bibliography: C.C.W. Taylor, The Atomists: Leucippus and Democritus (Toronto, 1999); G. Vlastos, Studies in Greek Philosophy, Vol. 1: the Presocratics, ed. D. W. Graham (Princeton, 1995); A. Barker, The Science of Harmonics in Classical Greece (Cambridge, 2007); A. Brancacci, “Democritus’ Mousika“, in eds. A. Brancacci and P.-M. Morel, Democritus: Science, the Arts, and the Care of the Soul (Leiden/Boston, 2007), pp. 181-206.