Plato, Pythagoras, and Stichometry

In Stichting Pythagoras
Pythagoras Foundation Newsletter. No.15. December 2010.

Plato, Pythagoras, and Stichometry

§1. Introduction

We know little of Plato the man and everything we need of Plato the philosopher. His name appears in the dialogues only three times, twice connected with Socrates’ trial and once with the day of his death. That is all. But to say that we know everything we need to know of Plato the philosopher is misleading because, according to Plato himself, philosophy is not to be qualified by anybody’s name. It deals with what is universal and eternal. One reason why Plato never appears as a character in his own dialogues is that he does not want to become, as it were, the subject-matter. This accords with the famous remark that Socrates can be refuted but not the logos.

One thing we learn from the dialogues is that, in a certain sense, philosophy has no need of Plato, it does not belong to him, it is not his possession. But it needs us, or rather we need it if we are concerned with ‘how a man should live’. Philosophy, according to Pythagoras, who coined the term, is love of wisdom and Plato is never tired of contrasting that love with the claim of the Sophists that they have wisdom and can dispense it, as if it were a commodity that they possess and can sell—if the price is right. The ‘knowledge’ of the Sophists brings them not only wealth but also fame and prestige, which, in turn, raises their market value.

Concern with ‘how a man should live’ is unfashionable these days, and a minor academic industry has been created around ‘Plato’, really around Platonism, which has nothing to do with that awkward question. Furthermore, most claimants to being Platonic scholars would deny that it was any of their concern; scholarship is not about how a man should live, but about what a philosopher said, wrote and meant—provided that it never meant that we should reflect on our lives, that we should live Socratic examined lives. But it does not deal with whether any of it is true.

Of course, modern scholars exist in an institutional framework which exerts strong influence upon them. They are paid by the institution, they are promoted (or not) by it, they acquire fame through it, and, if they are institutionally successful, they create through their students academic empires, political entities which support and promote, for the most part, their own orthodoxy. They are in constant danger of living in a form of Plato’s Cave—ironically indoctrinating their version of Platonism. To take only two examples of how this distorts whatever Plato himself understood, Raphael Demos openly confesses in his book on Plato that he did not understand mathematics, and Gregory Vlastos privately wrote that he understood nothing about music. One might have thought that these confessions would disqualify them from writing about Plato since it is clear that for him they were central to an understanding, central to philosophy. But no! Nor did it seem to occur to these worthies that they might actually learn some mathematics or study some music, subjects dear to Pythagoras and described in the seventh book of the Republic.

All these reflections were brought about by the appearance in the Texas academic journal, Apeiron, of an article by Jay Kennedy of the University of Manchester, with the title Plato’s Forms, Pythagorean Mathematics, and Stichometry.

There are three topics requiring attention. The first is the Manchester University publicity, the second is the claim of originality, and the third is the findings of the article. Since the present author is involved in the discussion, it will be written in the third person.

§2. Publicity

First, the publicity announces grandly that a “science historian” has “cracked ‘The Plato Code’–the long disputed secret messages hidden in the great philosopher’s writing.” This is pure hype—or rather impure hype. There is no ‘hidden code’ except in the sense that anyone who reads anything needs to know how to read. In a certain sense all writing is encoded—that is what is meant by ‘writing’ and to deal with it one needs skill in ‘reading’. Plato hid nothing, but succeeding thinkers (like Demos and Vlastos) forgot or never knew how to read. It would have been more significant if it had been declared that the existence and use of language is, in the old sense, a mystery, a mystery into which we must be initiated.

The ability to read, especially and in this case Plato, may be illustrated from the beginning of the Republic.

Its first word is ‘down-went-I’ (to the Peiraeus, it appears) which means that Socrates, who is recounting the dialogue, is now ‘back up’ (presumably in Athens). If he were still ‘down’ in the Peiraeus, he would have said ‘down-came-I’. There is no need to explore now the metaphorical significance of Athens and Peiraeus, but it is worth pointing out that the ‘down and up’ metaphor pervades the whole dialogue. The most obvious instance is the descent into the Cave (Book 7) and the laborious climb out of it. But if Socrates went ‘down’ at the beginning he comes back ‘up’ in the Myth of Er at the end of the dialogue, which explains the return. The dialogue, like all human knowledge, is circular.

The second word of the dialogue is ‘yesterday’. Now, first, if Socrates is telling us of what happened ‘yesterday’ he must be telling us ‘today’, that is to say, by the use of this one word Plato makes it clear that his dialogue is in an immediate and ever-present ‘now’. It is eternally contemporary. Second, because Socrates repeats yesterday’s conversation, we are presented with a double: there was the conversation yesterday and now there is its repetition today. In other words, we are presented with a ratio of 1:2, which would immediately suggest to any ‘musical man’ that we are involved with the octave, that is, that the dialogue must be understood musically. Third, Socrates follows the rule of the Pythagoreans that the first thing to do when waking is to go over in the mind the events of the previous day. The reader or hearer is alerted to the fact that the dialogue’s spirit is Pythagorean. There is here no ‘secret code’ only a sophisticated writing and reading ability. The Republic, like any of Plato’s dialogues, invites us to learn to read (or hear) with greater understanding. It is a self-instructing lesson in language.

This understanding of the first two words strongly influences the way the ensuing dialogue is understood. For example, if we understand that it is Pythagorean, we find the famous sentence in Book 1—‘This is no small thing we are discussing, but how a man should live’—more meaningful since it was well-known that to be a Pythagorean meant to have a way of life. But it also raises the question ‘How can arithmetic be a way of life?’ That needs to be pondered.

§3. Originality

Second, Kennedy’s claim to originality is mistaken. At least two writers had expounded the arithmetical and harmonical structure of Plato’s dialogues thirty years earlier. The story can best be told from the point of view of one of the two pioneers.

In 1959 John Bremer realized that the famous Divided Line of Book 6 of Plato’s Republic, a Line divided unequally, in fact divided the dialogue. By 1960 he had by rough count of lines strengthened his suspicion that the Divided Line divided the dialogue in what we call the Golden Section. He thus was drawn to see that the Republic had within itself its own reading instructions and began to use the Divided Line as an analytical tool for understanding the dialogue itself. But a more precise way of measuring the dialogue was needed and Bremer coupled with his discovery the simple fact that in the ancient world ‘publication’ meant a reading aloud, that is, the dialogue existed, not in space as on a printed page, but through time as it was recited. It was musical. The unit of speech in time is the syllable and so Bremer laboriously counted the syllables in the Republic (and most of the other dialogues), or, in other words, he used in a new way the ancient method of stichometry. He also discovered that the Republic takes twelve hours to recount.

The numbers confirmed his original intuition about the Golden Section and made it possible to identify accurately patterns and symmetries in the Republic, all based on the arithmetical counting of syllables. He never thought that the revealed patterns were simply a kind of ornamentation, a pleasurable addition to the content of the dialogue, a literary device, but rather that they were an essential, perhaps the essential, part of the dialogue. For example, the Golden Section was used because it is the well-known principle of growth and diminution, both in geometry and biology (see d’Arcy Thompson), and was therefore a self-referring creative principle.

Involved with other duties, Bremer could give little time to pursuing his discovery but by about 1980 he had prepared a book (published as On Plato’s Polity in 1984) outlining both the stichometric method and what it revealed about several of the dialogues. After the preparation, by chance in a bookstore, he came across a book that completed for him the arithmetical work that he had done. That book was The Pythagorean Plato by Ernest G. McClain and had been published in 1978.

From Plato’s statements in the various dialogues and on the basis of the actual numbers quoted in them, McClain established beyond any question that “Pythagorean harmonic theory is Plato’s ‘prelude to the song itself . . . the song itself that dialectics performs.’” He showed that the later dialogues—Republic, Timaeus, Critias, Statesman and Laws—embodied a systematic treatise on harmonics, on tuning theory.

To Bremer this came as an astounding revelation which opened up more possibilities for his arithmetical analysis. Through the good offices of Robert Brumbaugh (whose Plato’s Mathematical Imagination is a classic), Bremer and McClain were soon in touch and have collaborated ever since, sharing their work and thoughts. It turned out that McClain had published an earlier book The Myth of Invariance in 1976 with the intriguing and revealing subtitle of The Origin of the Gods, Mathematics and Music from the Rig Veda to Plato. This had established the use of harmonical allegories in Plato and other literatures.

While Bremer could take advantage of the profound musical knowledge of McClain, McClain was no longer restricted to the numbers that Plato actually quoted but could use the syllable count to extend his analysis of the harmonic structure in the dialogues. The two complemented each other in a most fruitful way.

In 1994 McClain published an introductory essay Musical Theory and Ancient Cosmology which confirmed and extended his findings about Plato. And in the issue No.169, Winter 2000 of Hermathena, A Trinity College Dublin Review, Bremer published an article introducing and extending his discoveries, Some Arithmetical Patterns in Plato’s Republic. In 2002, Bremer produced a detailed arithmetical-harmonical analysis of the Republic in Plato and the Founding of the Academy. (McClain drily observed that the Republic embodies a treatise on equal temperament). In 2005 Bremer’s study of a short dialogue the Ion appeared, with a translation and commentary; Plato’s Ion: Philosophy as Performance included a lengthy section on ‘Philosophy the Greatest Music’ and on Apollo’s Number (7776).

In the light of all these publications from 1976 onwards the priority for the discovery of the harmonical analysis of Plato’s dialogues must be accorded primarily to McClain, and the priority of the arithmetical and stichometric analysis to Bremer. This is not to denigrate the work of Kennedy, only to establish where the honor truly lies.

A further element in this history is that one of Bremer’s students, Maya Alapin, studied at Oxford but found it very difficult to find any supervisors to oversee her work. She carefully explained to many of the scholars who might have been suitable and willing to supervise her studies that she wanted to evaluate and enlarge the findings of Bremer. Although few were willing to help her, the outline she presented got passed around, not always being taken seriously, among the academic community and undoubtedly reached Manchester with or without acknowledgement. Alapin was awarded the MPhil degree in June 2007 and her thesis The Architectonic Structure of Plato’s Republic was duly made public. In the Introduction she writes:

It has been argued by John Bremer that there is a mathematical
and musical structure underlying the whole text of Republic, one
that is derived from the well-known Pythagorean ratios that
define the musical octave and the diatonic scale.

In Oxford Alapin was able to persuade Benjamin Weaver to create a software program, Panza, that would count the syllables of a Platonic text. Its use confirmed Bremer’s hand count of the length of the Republic at 180,000 syllables.

§4. The Kennedy article

First and foremost, whatever else may be said, an article analyzing the Platonic dialogues in mathematical and harmonic terms is to be welcomed and studied with great care, not least because it lies outside the conventional approaches to Plato. Furthermore this article is written by a scholar, Jay Kennedy, with some knowledge of and sympathy for both mathematics and music, unlike those mentioned in the Introduction, and therefore must be taken very seriously. The following remarks must be taken in the light of this overall commendation.

The question of priority is not important and one could wish that names and personalities could be ignored—as Plato wished. It is a pity that Kennedy either did not know or chose to suppress the earlier work of McClain and Bremer—their works being available in the Bodleian—but it really does not matter.

While Kennedy clearly understands something of both mathematics and music, it is not clear that he understands them as they were understood in the ancient world. For example, it is not evident that, in music, he knows that Plato worked with four different tuning systems—Just, tempered, Pythagorean, and Archytan (corresponding to the four Platonic cities of Atlantis, Callipolis, Athens, and Magnesia)—and that, in mathematics, he was concerned with what the modern world calls Diophantine approximation. Turning Greek mathematics into modern terms is dangerous and misleading, as evidenced in the work of Thomas Little Heath for he assumes that a ratio is a number (which it never was, for it was a relation between numbers), and that one is a number (rather than the origin of all numbers).

Kennedy rightly emphasizes the importance of the number twelve—and certainly Plato focuses on the problem of dividing a cyclic octave into twelve equal parts, an impossible task without the use of irrationals—but he seems to ignore the significance of the number ten (so important to the Pythagoreans) and its relation to the Forms. There is in Plato a continuing ‘dialogue’, as it were, between the numbers ten and twelve which are reconciled in the number 60, the root of the Babylonian sexagesimal system.

In this connection, the title of the article promises more than it delivers for it has little to say about the Platonic Forms. A connection must be made between the arithmetical and harmonical structure of the dialogues and the Forms. This Kennedy does not do—does not even allude to it. Nor does he appreciate the way in which music exists through time; he would if he saw the dialogues not as seen, as static written texts occupying space, but as heard, as dynamic sounded sequences in time; more generally as processes not products.

The stated intent of Kennedy’s article is “to corroborate the view of Aristotle and other members of the early Academy that (Plato’s philosophy) was fundamentally Pythagorean.” In a purely gross and verbal sense, ‘we need no ghost come from the grave to tell us this’, but, in a more exact sense, it would have been good to know precisely what it would have meant to call Plato a Pythagorean, and if he would have accepted the description. The name of Pythagoras occurs only once in all the dialogues and the Pythagoreans are mentioned twice, all three occurrences in the Republic. This is the same number as Plato’s name appears in all the dialogues, the paucity being attributable to the reason mentioned in the Introduction. The confusion stems from the assumption that being a Pythagorean meant subscribing to an orthodoxy, whereas it actually meant following a way of life (as Plato states in the Republic, St.600b.2). Aristotle, mistakenly and for his own purposes, thinks only of a doctrine and never of the way.

It is to be hoped that Kennedy will pursue his line of thought but with a more thorough search of the literature, bearing in mind that academic philosophy is a subject-matter whereas Platonic and Pythagorean philosophy is a way of life.