The Nuptial Number Revisited

The so-called 'Nuptial Number' is by far Plato's most intractable mathematical problem. It is so dense and so obscure that many commentators, both ancient and modern, have decided that it has no rational solution but is instead an item of satire, not to be taken seriously. Others - including the present author (an admitted obscurantist) - see it as one of the great keys of the Platonic canon, if only it is understood correctly. Like the impossibly obscure formulae of the alchemists, it is the key to the Mysteries. 

It is not the purpose of this post to offer any sort of detailed solution to the 'problem' posed by this most difficult of passages from Plato's Republic; rather its limited purpose is simply to present the passage in question in a clear, accurate translation usefully divided into its component parts. The Greek text is impossibly convoluted and many of the words used are both rare and esoteric. No translation can save the passage from what seems to be deliberate obfuscation on the part of the author. But much of it can be disentangled and made clearer by some explanatory additions and a careful break down of its parts. 

What the infamous 'number' might be and what its significance might be is another matter, but the drift of the passage in itself  - leaving aside the formula of the number - can be straightened out into a largely lucid account. That is what is attempted below. The rendering is based upon those of Benjamin Jowett and James Adam (with copious references to the lexicon of Liddel & Scott. We begin at Republic 545D:

THE PROBLEM: WHAT CAUSES STATES TO DECLINE?

“Come, then,” I said, “and let us try to discern the way a timocracy will develop from of an aristocracy. Or is it the simple and unvarying law that in every form of government disturbance begins among the ruling class itself, when sedition arises among it, but as long as it is at one with itself, disturbance will not occur?”

“Yes, that is so.”

“How, then, Glaucon,” I said, “will disturbance arise in our ideal city, and how will our auxiliaries and rulers fall out and be at odds with one another and with themselves?

INVOCATION TO THE MUSES

Shall we invoke the Muses as Homer does in order to tell us “‘how faction first fell upon them,’” and shall we, like a tragic poet, picture the Muses speaking in an elevated style, as if they are speaking seriously when in fact they are playing with us and teasing us like one child teases another?

“How?”

“In some such fashion as follows:

THE WORDS OF THE MUSES

[Having been invoked in the Homeric manner - albeit ambiguously - the Muse now speaks, and it is through this quasi-Homeric Muse that we are given the Nuptial Number. It is an open question as to how seriously we should take this entire construction, although let us note that Plato uses this "Muse" in other contexts in other dialogues. Many of the 'Platonic myths' are introduced in this way.]

ALL THINGS PASS

It is indeed difficult for a state ideally constituted as we have described to be shaken and disturbed; but since for everything that is born there is also a time of destruction, even a constitution such as ours will not last forever, but it shall surely be dissolved eventually.

This is the manner of its dissolution:

THINGS MOVE IN CYCLES

Not only for plants that grow from the earth but also for animals that live upon it there is a cycle of fertility and barrenness of soul and body as often as the revolutions of their orbs come full circle, in short cycles for the short-lived things and long cycles for long-lived things.

BEGETTING CHILDREN OUT OF SEASON

Concerning human beings, there are also laws of fecund birth and of infertility, and there will come a time when these laws will escape the men you have bred to be the rulers of your city. For all their wisdom, and combining calculation with observation, they will beget children out of season.

A PERIOD COMPREHENDED BY A NUMBER

Now for divine creatures there is a period comprehended by a number that is final and perfect, but for a mortal the number is the first in which multiplications of root by square - when they have attained three distances, with four limits, of that which makes like and unlike and waxes and wanes - have rendered all things commensurable with one another.

THE BASE PRODUCES TWO HARMONIES

The base of this, containing the ratio of four to three, yoked with five, produces two harmonies when increased three times.

1. One of them is equal an equal number of times, so many times a hundred.

2. The other is equal length one way but oblong

- one dimension of a hundred numbers determined by the rational diameters of five diminished by one in each case, or of the irrational by two; the other dimension of a hundred cubes of three.

LORD OF BETTER AND WORSE BIRTHS

The sum of these - this entire geometrical number, a number measuring the earth - is lord of better and worse births.

THE NEGLECTFUL GUARDIANS

When the guardians of your ideal state neglect this and marry brides to bridegrooms out of season children of ill-nature and ill-fortune will be born.

NEGLECT OF THE MUSES

The best of their predecessors will indeed make rulers but these offspring, being unworthy, when they have succeeded to their fathers' offices of power, will begin to neglect us muses, though they are our guardians, and will pay too little heed to music, and then to gymnastics, so that the children will deteriorate and grow up without us.

THE METALLIC RACES

And the rulers who come after them will have little of the guardian in them for testing Hesiod's races and your own - races of gold and silver and copper and iron. And iron will be mixed with silver, and copper will be mixed with gold, and this will engender unlikeness within them, and an unevenness which is disharmonious, which things always create war and enmity wherever they are found. This surely is the pedigree of sedition, wherever it arises. As Homer says, "Of this lineage, look you!" (Iliad 6:211)

CONCLUSION

[Here the Muses finish and we return to the conversation between the interlocutors of the dialogue, namely Socrates and Glaucon.]

"And quite right too," said Glaucon. "We affirm what the Muses say as correct."

"Indeed," I said, "because they are Muses."

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Yours,

Harper McAlpine Black

Roger Sworder the Platonist

Science & Religion in Archaic Greece

Amidst the madness and cacophony of late post-industrial modernity there are, hidden here and there, still a few quiet voices of perennial sanity. One of them is Roger Sworder the Platonist. Since retiring from a long and distinguished career as a University lecturer – known for his inspiring and compelling lectures – he has been especially productive, publishing numerous books of essays on Platonic and related themes. He stands among the company of a new breed of Platonic scholar who reject the dry, purely textual and theoretical approach to Plato and ancient wisdom traditions promoted in the Western academy over the last few centuries. Instead, like such scholars as Arthur Versluis, Pierre Hadot and Peter Kingsley – and following such maligned advocates as Thomas Taylor in former times - he sees the Greek heritage as a sapiental tradition, a spiritual tradition, that was and is central to Western civilization. 

In Dr. Sworder’s case he is a graduate of Oxford University, but he has few kind things to say about that august institution. He argues that the way the Greek philosophical tradition has been taught in such institutions of higher learning amounts, in fact, to a type of anti-philosophy, the very opposite to a tradition of spiritual wisdom. Indeed, he supposes that the strangulation and deformation of the ‘Classics’ in such institutions has been a deliberate device in the construction of the atheistic wasteland of Western modernity. In part, this is why he has spent his entire career teaching and writing in the remote backwaters of regional Australia. It was in a small, insignificant rural college that he was able to construct and teach a viable programme of studies outside of the strictures of the academic mainstream. This programme took the form of a degree course entitled ‘Studies in Western Traditions’ and then, later, in the wake of the endless amalgamations and restructurings to which Australian universities are prone, in a major programme entitled ‘Philosophy & Religious Studies’. His special expertise in all incarnations of those programmes was the Greeks – Plato, the Presocratics and Homer – and also the English Romantics. Away from the stuffy conventionalism of Oxford he was able to teach as he saw fit for over thirty years. Now he devotes his days to a semi-monastic contemplative existence writing and publishing.

A strident critic of corrosive technologies, he is happily internet-free, has never sat at a computer, composes his work in long-hand and indulges in neither email nor mobile phone. This is not just a stubborn Luddite posture, though; he has a genuine and deeply Platonic concern for the dehumanizing impact of technology, especially on meaningful human work. His deepest fear is that mankind is busily constructing what he calls a “toy store” of gadgets for itself without any notion of why or to what end. For many semesters his flagship subject was entitled ‘Philosophy of Work and Art’, delivered to students of both the Bachelor of Arts and the Bachelor of Fine Arts. Citing the teachings of the Republic – as also the Bhagavad Gita from the Hindoo Tradition – he sees meaningful work as the centerpiece of the human estate and the degradation of work as the most damning feature of industrial modernity. Further still, in the tradition of Thomas Taylor, he is a careful and precise critic of the methodical excesses of the modern sciences that underpin technology, the great shibboleth of our times. His disdain for the vulgarity of techno-modernity is legendary among his friends. A non-driver, he curses the age of the automobile. He has a Pythagorean devotion to fine music (he is married to the Chinese concert pianist, Nan Chien) and a corresponding hatred of noise. Nothing rouses his ire like the infernal racket of machines.

Mining, Metallurgy & the Meaning of Life

A deliberate anonymity is an important feature of Dr. Sworder’s entire manner. He lives in a remote corner of the world, communicates little, and writes his books without fanfare or efforts at self-promotion. He trusts that his work will reach those few who might profit from reading it. He does not indulge in the egoic antics of some and is wary of academic careerists. More to the point, he stays aloof from the New Age fads that occupy the borderlands of the ‘spirituality’ movement. He is neither a shaman nor a Sufi nor indeed an upper-case T ‘Traditionalist’. His writing is simple, clear, sparse and unadorned. He feels no need to show off his skills in Greek and Latin nor the great depth of his reading. His style is understated and modest but readers can be assured that it rests on the solid foundations of a lifetime’s earnest and dedicated study. He avoids verbosity. He is not out to match wits with self-important professors or to dazzle readers with his references. Nor is he building a personality cult. He is offering a sober, penetrating reading of Plato and the ancients relevant to our times and to the impasse in which we find ourselves based on hard-won observation. He receives little credit or acclaim, yet he is – without doubt – one of the foremost Platonists writing in English today. 

A Contrary History of the West & Other Essays

The cornerstone of Dr. Sworder’s ouvre is his work on the father of occidental ontology, Parmenides of Elea, and the relation of that thinker to Plato. He offers his own translation of the famous (fragmentary) poem of Parmenides and an interpretation of its symbolism and implicit cosmology. Although Parmenides is counted as the great metaphysician of the Western tradition, Sworder proposes that the proper way to approach his metaphysics is through cosmology. He reconstructs the parallelisms of this cosmology, seeing the goddess Aphrodite (and her star, the planet Venus) as central to the Parmenidean vision. It is Aphrodite (Venus) who is sentinel to the Palace of Night. Moreover, in a series of simple and elegant correspondences, Sworder shows how this reading of Parmenides demonstrates how the Delphic oracle worked in principle and how it offers a solution to Plato’s most intractable mathematical problem, the Nuptial Number. This was published in a private edition many years prior to the now best-selling venture of Peter Kingsley in The Dark Places of Wisdom and has been republished. For those who are wary of Kingsley’s neo-Shamanic and anti-Platonic primitivism – a journey into a pre-rational darkness - Roger Sworder offers a different account, one that unites Parmenides and Plato in the same tradition. 

Mathematical Plato

In the opening essay of his more recent book Mathematical Plato, he provides an altogether luminous reading of Plato’s most troublesome dialogue, called the Parmenides, in which he offers a simple way to resolve apparent difficulties – from a Parmenidean point of view – arising out of Socrates’ pet theory, the Theory of Forms. Sworder sees Plato as using the monism of Parmenides to resolve the apparent dualities of that Theory and in the process offer a vision of an ‘optimal’ world, the best of all possible worlds. Sworder’s Platonism, that is to say, is not in any sense world-denying.

It is worth quoting the final paragraphs of that essay, since they provide a capsule statement of Sworderean Platonism:

Plato’s theory of ideas expresses that view of the world around us where everything is very beautiful. The myths of the Phaedo and Republic, Phaedrus and Statesman spring from this same visionary power. This is the world of Parmenides’ chariot ride to the palace of a Goddess who reveals all things to him; the world of the Iliad and the Odyssey. Divine and mortal freely mix. Everything is well made and performs at the limit of its potential; our everyday activities are archetypal. It is perhaps a greater injustice than Socrates suffered that Plato should ever be considered a Utopian idealist who despised the world.

Fully to understand how this world is the most perfect possible realization of the fullest totality of the most exquisite ideas is a Her- culean education. The deepest seclusion is needed to complete a thorough study of arithmetic, geometry, astronomy, music and dia- lectic. After these come their applications to the natural sciences. These studies are typically pursued through early adulthood to middle age. That done, the meaning of our human life emerges as a vision in which no further parting is possible between the absolute and the relative, the eternal and the temporal. There is apocatastasis. This is the goal of Plato’s theory.

The present writer must confess to being a student of Dr. Sworder and to regarding him as a mentor. 

The works of Roger Sworder are available through Sophia Perennis Press. 

The Romantic Attack on Modern Science in England & America

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Yours,

Harper McAlpine Black