Pythagoreanism, the Divine, and the Nature of Eternity

[I compiled this paper in December 2017 to provide background detail for a speculative discussion of ancient patterns of thought outside Greece, and in particular in connection with thought in Britain before the arrival of the Roman legions. This question is worth looking at since we are told by a number of ancient writers that there were certain resemblances between the ideas and beliefs held by the priests in the British Isles, and those associated with the followers of Pythagoras.

That was a very specific reason for compiling this paper, but the compilation of the details of Pythagorean doctrine and its resemblances, and the discussion of the likely philosophical background to Pythagorean doctrine,  is useful in itself. So I've extracted that part of the paper, and re-edited it to stand alone, along with a little material that was not in the original]. 

Abstract: This paper explores the idea that there is a connection between some core Pythagorean mathematical and geometrical concerns, and ideas of divinity and Eternity.  On the basis of a close examination of Pythagorean ideas in the 1^st millennium, for which we have extensive documentation, It is suggested that this connection is a logical one.

Key words: Pythagoras, Philosophy, Religion, Number, Mathematics

1 The Longevity of Ideas

We often underestimate the longevity of patterns of ideas. Sometimes when they are linked to a religious or theological structure, they can have a very long existence. Though much of modern knowledge about the physical world has been developed since the European Enlightenment, there are still ideas around which have persisted with very little change, since the first millennium B.C.E. Hinduism is still much as it was for example, as is Buddhism. Later religions such as Christianity, built as it was on the Old Testament, preserves many aspects of Hebrew ideas [Christ is made to paraphrase YHWH’s statement in the Old Testament that he is ‘first and last, and beside him there is no other god’, by characterising the divine as the ‘alpha and the omega’].

In short, there are still religious ideas and formulations around in the world, and contained in the human mind, which are more than two and a half millennia old. And in some cases, much older than that. Languages and peoples may change, but ideas are sometimes much slower to change, and may survive alteration of language, people, and material culture.

This paper explores a hypothesis: the hypothesis that some ideas which we habitually consider to be around two and a half millennia old, are in fact much older than that. In the west, these ideas find powerful expression in Pythagoreanism, written about by both Plato and the later Neoplatonists. Looked at in the Greek context alone, this body of ideas extends over nearly eleven hundred years (if a floruit of the mid sixth century BCE for Pythagoras is correct), until the closure of the philosophical schools in 529 CE.

So how old is philosophy? Classicists are ready to be very specific about the matter. Their view (which became orthodoxy during the European Enlightenment of the eighteenth century) is that the Greeks pioneered philosophy from the seventh and sixth centuries B.C.E. onwards, culminating in the intellectual enlightenment of the 5th and 4th centuries B.C.E.

Yet as late as the 2nd century of the Common Era, the Christian writer Clement of Alexandria said that:

Philosophy..., with all its blessed advantages to man, flourished long ages ago among the barbarians, diffusing its light among the gentiles, and eventually penetrated into Greece.

Note that Clement places the Greeks not at the beginnings of philosophical thought, but as the late inheritors of a practice which had flourished for long ages, and among those who did not speak Greek.

Clement goes on to be more specific, and says of philosophy that:

Its hierophants were the prophets among the Egyptians, the Chaldeans among the Assyrians, the Druids among the Galatians, the Sramanas of the Bactrians, and the philosophers of the Celts, the Magi among the Persians who announced beforehand the birth of the Saviour, being led by a star till they arrived in the land of Judaea, and among the Indians the Gymnosophists, and other philosophers of barbarous nations.

— Clement of Alexandria, Stromata 1.15.71 (ed. Colon. 1688 p. 305, A, B).

So there is the possibility that there is another narrative which we can reconstruct. 

2 Pythagoreanism in 1^st Millennium Britain

We have Greek and Roman sources for the supposed origins of Pythagorean modes of thought. These point in different directions. We have the story that Pythagoras was present at the fall of Babylon in 539 BCE, and he is also supposed to have spent some time in Egypt (around twenty years), learning from the priestly cults. On the other hand, we have information about the beliefs of the Gaulish priests from the mid-first century B.C.E., in the wake of Julius Caesar’s campaigns in north western Europe. Caesar described the Gauls in his Commentarii de Bello Gallico, [The Gallic War], book VI.

According to Caesar, the Gaulish priests were concerned with "divine worship, the due performance of sacrifices, private or public, and the interpretation of ritual questions." He also said that they played an important part in Gaulish society, being one of the two respected classes, the other being the equites (the Roman name for ‘’knights - members of a privileged class able to provide and equip horsemen). They also functioned as judges in disputes. Among other interesting details, Caesar also said that they met annually at a sacred place in the region occupied by the Carnute tribe in Gaul (possibly Chartres), and that Britain was the home of priestly study. Caesar also said that many young men were trained as priests, during which time they had to learn large amounts of lore by heart. 

Metempsychosis was the principal point of their doctrine: “the main object of all education is, in their opinion, to imbue their scholars with a firm belief in the indestructibility of the human soul, which, according to their belief, merely passes at death from one tenement to another; for by such doctrine alone, they say, which robs death of all its terrors, can the highest form of human courage be developed”.

He also tells us that they were concerned with "the stars and their movements, the size of the cosmos and the earth, the world of nature, and the powers of deities". So the components of their religious cult involved the study of astronomy, cosmology, natural philosophy, and theology.

Alexander Polyhistor, in a passage preserved by Eusebius, described the Gaulish priests as philosophers, and explicitly called them ‘Pythagorean’ on account of their understanding of reality. He wrote that "The Pythagorean doctrine prevails among the Gauls' teaching that the souls of men are immortal, and that after a fixed number of years they will enter into another body."

Diodorus Siculus, writing in 36 BCE, also said that the Gaulish priesthood followed "the Pythagorean doctrine", that souls "are immortal, and after a prescribed number of years they commence a new life in a new body."

There are other descriptive references to the Gauls and their religion from antiquity, but it is not necessary to review all of them here. These are the main evidential details we have for the presence of Pythagorean ideas in Gaul and in Britain in the last two centuries of the 1^st Millennium BCE. It is likely that both Polyhistor’s account and the account of Diodorus Siculus drew on the source used by Caesar.

3.The Principal Sources for Pythagoreanism

The preceding descriptions are usually all that is mentioned when religion in Gaul and in Britain before the arrival of the Romans is discussed. This is because we do not have written records from Gaul or Britain from earlier times. And so this is where historical discussion usually stops. The rest of the story of these cultures becomes a matter for archaeological investigation.

However, we need not stop here, looking at nothing. Much of what we know about the other philosophical details of the Pythagoreans is quite extensive, if not always consistent across the range of sources.  There is a life of Pythagoras by Iamblichus, and another by his pupil Porphyry. A life of Pythagoras by Diogenes Laertius also contains useful information. Plato and the later Platonists wrote in detail about Pythagorean doctrine, if not always being explicit that they were referencing his ideas. 

Plato is the best place to start. He had the concept of an inner and outer knowledge, which reflects something of a priestly understanding of both teaching and of reality. He referred to these grades of knowledge as ta eso and ta exo. Which means that teaching operated at two levels – the exoteric and public level, and another which was esoteric in nature.

Esoteric knowledge is by definition obscure, and/or difficult to understand. Which is what the story of the prisoners in the cave in Plato’s Republic is all about. They see the shadows of reality on the wall before them, but not the reality itself. When they are released with suddenness, their reason is deranged by the experience. Instead they should have been released gradually, being shown details of reality first, without the whole of the shocking truth of reality being given to them all at once.  Plato was engaged with both exoteric and esoteric understandings of knowledge, but mostly what he tells us about is an esoteric doctrine, which explains what is hidden and obscure, and relates to the gods, and what is divine. As one might expect, the rules for the gods are different.

4 The Core of Pythagorean Doctrine

In the Timaeus Plato refers to a principle of wholes, or totalities. It is later mentioned by the Neoplatonist Porphyry as a Pythagorean doctrine, and Pythagoras is supposed to have learned of it in a lecture in Babylon, after the fall of the city to the Persians in 539 BCE. The doctrine is of course, very much older. It can be detected in the Iliad, in Bk 18, where Hephaestus makes objects which, on account of their nature, can pass into the counsel of the gods, and return. The principle might, as Porphyry suggests, have been brought back to the west by Pythagoras after his spell in the east, or it may already have been part of a body of ideas already well established in Italy and in Greece. The principle might be simply put, as ‘things which are total participate in totality’, in the same way that Plato declared that ‘greatness is participation in the great.’ But it is so much more important than a statement that wholes conjoin with one another. It is the essence of the ascent from image to image to an apprehension of the Good which Plato refers to in both the Timaeus and the Republic.

Each of these images must represent or embody an aspect of what Plato referred to as ‘the Good’. Each of the images must allow the supplicant to pass from one to the other via their essential identity. What varies between them is the degree of their participation in the Good.

Plato is very clear that the viewer of the images must be able to pass along the chain of images in either direction. The chain of images is not therefore purely about gaining an understanding of the Good (meaning the divine, or Being itself), either in reality or figuratively. Passage through the chain of images is about both the transcendence of images or forms, and about the descent of Being into the world of generation, as a generative power. The images are constructed in the way they are in order to reduplicate and re-energise the power and presence of divine Being in the human world. For man, this might be seen as an act of worship or observance of what is holy, but it can also be understood also as a form of theurgy, even if the technical term post-dates classical Athens by several centuries.

In the Timaeus [30a-b], Plato speaks through Timaeus, saying:

For God desired that, so far as possible, all things should be good and nothing evil; wherefore, when He took over all that was visible, seeing that it was not in a state of rest but in a state of discordant and disorderly motion, He brought it into order out of disorder, deeming that the former state is in all ways better than the latter. For Him who is most good it neither was nor is permissible to perform any action save what is most fair. As He reflected, therefore, He perceived that of such creature as are by nature visible, none that is irrational will be fairer, comparing wholes with wholes, than the rational….

Plato, in using the phrase ‘comparing wholes with wholes’, is referring to the principle of wholes and totalities mentioned in Porphyry’s account of Pythagoras.

Pythagoras is said by Porphyry to have associated with the ‘other Chaldeans,’ after he mentions his conferring with the king of Arabia. The current academic view is that the Chaldean dynasties were essentially Arab dynasties, and that they were in control of Babylon at this time.  This helps to confirm the reliability of some of the detail in this important passage, written so long after the lifetime of Pythagoras. 

What did Pythagoras take from his long sojourn in Egypt, and the near-east? Is his doctrine like Plato’s? The point of the doctrine of wholes and totalities, is to establish connection between the divine world and secular reality. Porphyry’s account tells us that:

He cultivated philosophy, the scope of which is to free the mind implanted within us from the impediments and fetters within which it is confined; without whose freedom none can learn anything sound or true, or perceive the unsoundedness in the operation of sense. Pythagoras thought that mind alone sees and hears, while all the rest are blind and deaf. The purified mind should be applied to the discovery of beneficial things, which can be effected by, certain artificial ways, which by degrees induce it to the contemplation of eternal and incorporeal things, which never vary. This orderliness of perception should begin from consideration of the most minute things, lest by any change the mind should be jarred and withdraw itself, through the failure of continuousness in its subject-matter.

To summarise: the principle of wholes can be understood as a logical modality which connects the world of the mundane with transcendent reality. The definition of transcendent reality in Plato (articulated by Socrates) is that it is a place beyond shape, form, size, etc., and occupies no place on earth. It is however the place where knowledge has its reality (the ‘eternal and incorporeal things’ mentioned by Pythagoras). Connection with transcendent reality is possible by the likenesses to the transcendent which have existence on earth, such as things which are complete and whole, which therefore participate in the completeness and wholeness of the transcendent reality. Completeness and wholeness require (in the world of the mundane) delineation and limits, and so the limits and the extremes of things are also things which participate in transcendent reality.

The principle of ascent to the ‘eternal and incorporeal things’ in the doctrines of both Plato and Pythagoras, is entirely a mental process, which does not involve any of the senses. It proceeds via chains of similitudes, both up and down, as a sequence of orderly perceptions. The goal is a form of communion with that which never varies, and which is always one and unchanging, as Plato tells us in the Sophist. The return from the communion with the Good delivers beneficial things, because the Good is the source of all knowledge.

5 Diogenes Laertius on Pythagoreanism

Diogenes Laertius is generally not regarded as a great historian of the philosophy of the ancient world, but his Lives of the Philosophers is the only general account which survives from antiquity. We get snippets from elsewhere, but not the comprehensive sweep that he gives. He does not always have good materials, or understand them well. But with his writing on Pythagoras, we get something different. He is working with some very good materials indeed. His date (actually quite uncertain) may be contemporary with the Neoplatonists who also wrote about Pythagoras, and possibly he is using the same now long vanished materials, since he reproduces the same sort of inconsistencies of detail which appear in Iamblichus’s Life of Pythagoras. These inconsistencies, mainly concerning religious observances, may be explained by the fact that the core of Pythagoras’s doctrine isn’t about these things at all, but about an agrapha, or ‘unwritten doctrine’, revolving around deeper matters.

As already mentioned, it was a popular opinion in antiquity that Pythagoras did not write any books – “There are some who insist, absurdly enough, that Pythagoras left no writings whatever” [D.L., Book VIII, 6], however Diogenes says that ‘Heraclitus, the physicist, almost shouts in our ear, “Pythagoras, son of Mnesarchus, practised inquiry beyond all other men, and in this selection of his writings made himself a wisdom of his own, showing much learning, but poor workmanship”’. Heraclitus therefore seems to recognise the disparate origins of the material used by Pythagoras (in his book On Nature), and feels that it has not been worked properly. Diogenes tells us that Pythagoras wrote three books altogether, which were (since we no longer have them): On Education, On Statesmanship, and On Nature. Other works were also attributed to him [D.L., Bk VIII 7].

Diogenes appears to have had access to the three Pythagorean texts, or extracts from them, or perhaps from epitomes of them, since he talks about the contents [D.L., Bk VIII 9-10]. He says that Pythagoras was understood to be the first to speak of the idea of metempsychosis – he declared that “the soul, bound now in this creature, now in that, thus goes on a round ordained of necessity”. D. L. says that ‘so greatly was he admired that his disciples used to be called “prophets to declare the voice of God” [D.L., Bk VIII 14].

The books seem to have been kept secret, since Diogenes says that “Down to the time of Philolaus it was not possible to acquire knowledge of any Pythagorean doctrine” until Philolaus “brought out those three celebrated books.” Diogenes says that Plato sent a hundred minas in order to purchase these texts [D.L. Bk VIII 15]. He cites Aristoxenus in the tenth book of his Rules of Pedagogy “where we are …. told that one of the school, Xenophilus by name, asked by someone how he could best educate his son, replied, “By making him the citizen of a well-governed state.”’ This is of course the clearest anticipation of Plato’s interest in education.

Diogenes relates some details, not always in agreement with each other, of the religious nature of Pythagoras' philosophy: “He used to practise divination by sounds or voices, and by auguries, never by burnt offerings, beyond frankincense. The offerings he made were always inanimate; though some say that he would offer cocks, sucking goats and porkers, as they are called, but lambs never. However, Aristoxenus has it that he consented to the eating of all other animals, and only abstained from ploughing oxen and rams” [D.L., Bk VIII 20]. Diogenes relates later that ‘Apollodorus the calculator’ says “he offered a sacrifice of oxen on finding that in a right-angled triangle the square on the hypotenuse is equal to the squares on the sides containing the right angle”.

There was therefore some uncertainty in antiquity about exactly what the religious practice of Pythagoras was – it may not have been consistent in its nature, and it follows that it is possible that some of the practices attributed to Pythagoras, (vegetarianism, avoidance of killing animals, the avoidance of beans, etc.) are not in themselves of essential importance to Pythagorean doctrine, but only seemed so to compilers and commentators in late antiquity.

If we look at some further statements by Diogenies we can guess what the important things in Pythagorean doctrine are. Diogenes says that Pythagoras advised his disciples to say to themselves when entering their own doors: ‘Where did I trespass? What did I achieve? And unfulfilled what duties did I leave?’ [D.L., Bk VIII 22]. This indicates (among other things) the importance of the threshold or limit to Pythagoras.

Pythagoras also urged that the memory be trained. This was also extremely important to Plato, and he regarded the invention of letters to have been a disaster on the grounds that they impaired the training of the memory through making its importance less clear. There were in any case already people in Greece who held large parts of the Homeric poems in memory, since the poems were not committed to writing until the time of Peisistratus (some time after he first became tyrant of Athens in 560 BCE). Memory seems to have been cultivated in Egypt, and was certainly practised (and discussed) in late antiquity in various parts of the Roman Empire (Cicero mentions it, and it surfaces in the work of St. Augustine).

Pythagoras also said that men should sing to the lyre and by hymns to show due gratitude to gods and to good men. He bade men “to honour gods before demi-gods, heroes before men, and first among men their parents”. The principal image here is the gods, who are more important than the demi-gods, in terms of their claim on our worship and honour. Heroes stand in the same relation before men, and our parents stand in the same relation to us. He amplifies the importance of this metaphorical perspective, by saying that men should ‘honour their elders, on the principle that precedence in time gives a greater title to respect; for as in the world sunrise comes before sunset, so in human life the beginning before the end, and in all organic life birth precedes death’ [D. L., Bk VIII 22-4].

At one level this kind of metaphor-making looks trivial, which is one of the reasons why little has been made of these passages. However. Pythagoras is setting up oppositions between extremes within defined classes (Gods and demi-gods, who are immortal, Heroes and men, who are mortal, etc.), and making a comparison between them. He is also establishing a line of connection between them. He isn’t just comparing one image with another, he has created chains of images, with one end of the chain representing the extreme of reality (the Gods), and we stand at the other extreme.

The image of ourselves and our parents might be taken to suggest a parallel with the relation between Gods and demigods. In terms of the relationships implied in the image, the familial image can be understood as a copy of sorts, more or less imperfect, of the relationships between Gods and demigods. We are of course familiar with the Greek Gods and their shocking personal relations with each other, which often suggest an earthly and dysfunctional extended family.

Like Plato, Pythagoras had an agrapha, since some Pythagoreans “used to say that not all his doctrines were for all men to hear” – which is perhaps why it was so difficult to acquire knowledge of Pythagorean doctrine until the indiscretion of Philolaus [D.L., VIII 15-6]. Diogenes authority for this is the tenth book of Aristoxenus’ Rules of Pedagogy. Diogenes draws details of the Pythagorean philosophy from another lost author – Alexander, author of Successions of Philosophers, who claimed to find the following in the Pythagorean memoirs:

The principle of all things is the monad or unit; arising from this monad the undefined dyad or two serve as material substratum to the monad, which is cause... [D.L., VIII 25].

This is very like the conception of the Neoplatonists, who argued that in order that the good should remain untainted with generation and change, a copy came into being, which did participate in creation:

from the monad and the undefined dyad [the ‘undefined dyad’ may also be translated as ‘unlimited dyad’, or ‘unbounded dyad’ (the Greek term is ‘aoriston’) spring numbers; from numbers, points; from points, lines; from lines, plane figures; from plane figures, solid figures; from solid figures, sensible bodies, the elements of which are four, fire, water, earth and air. These elements interchange and turn into one another completely, and combine to produce a universe animate, intelligent, spherical, with the earth at its centre… [D.L., VIII 25].

Once again we have a chain of images: the monad and the undefined dyad, numbers, points, lines, plane figures, solid figures, sensible bodies, the four elements. Note that we don’t have the monad included in his sequence – the first image in the sequence is the monad and the undefined dyad, which “serves as material substratum to the monad, which is cause…”

Each of these conceptions is an imaging of the properties of the monad, one leading to the other, increasing in definition, attributes and properties until we reach the sensible bodies with the properties of fire, water, earth and air, all of which can interchange into one another completely. This chain represents an order of generation, rather than an order of perception.

That each of the sensible bodies can interchange into one another completely is a corollary of the fact that they arise from the undefined dyad. They are differentiations of the undefined dyad, and their fundamental identity resides there. Thus, fire may stand metaphorically for water, earth for air, and so on. The interchange occurs with reference to the monad and undefined dyad, since the combination of the monad and undefined dyad is the fount of all cause, generation and change.

The monad itself is an image, but can have no definition beyond the ‘One’. The numbers did not arise from the monad itself: for that to happen, something else was necessary. That he does not say that numbers arise from the monad is an important clue towards understanding that not only is Pythagoras speaking in terms of images, but that these images are related to an account of another level of reality, referred to, but not articulated. This level of reality is entirely without rational form, and which has an esoteric nature. Such an account of reality also informs Hesiod’s version of the creation, in which there is an ultimate reality which does not conform to the categories of our understanding (i.e., ‘chaos’).

In addition, there is a strong ethical element to Pythagoras’ life and philosophy, and it is clearly associated with the properties of the threshold, where one thing conjoins with, or turns into another. It would not be unreasonable to describe Pythagoras’ philosophy as theurgic in nature, since he is concerned that man should control his own destiny, and not trust to the gods alone. He also laid down precepts for religious practice and religious discipline, meaning that there was understood to be an efficacious ritual element in the transformation of the human soul.

For example, Pythagoras urged that victims ought not to be brought for sacrifice to the gods, and that worship should be conducted “only at the altar unstained with blood”. In addition, he stipulated that the gods should not be called to witness, “man’s duty being rather to strive to make his own word carry conviction”. He also said that men should avoid excess of flesh, and that they should respect all divination. Abstention from beans was recommended “because they are flatulent and partake most of the breath of life; and besides, it is better for the stomach if they are not taken, and this again will make our dreams in sleep smooth and untroubled” [D.L., Bk VIII 23-4].

That they ‘partake most of the breath of life’ is an objection to beans seems a little odd, unless by this Pythagoras is indicating that the target of the transformation of the individual through discipline, ritual and understanding is a condition which does not partake of the breath of life. We recall that Socrates in his final moments asked that a cock be sacrificed to Asclepius, which was to mark a return to health – in this case a healing from the trials and tribulations of life.

Pythagoras seems to have shared this view of earthly life. In vol 2, Bk. VII on Zeno (333-261 BCE, of Citium in Cyprus, a city which ‘had received Phoenician settlers’), we are told by Diogenes that the author Hecato, and also Apollonius of Tyre, in his first book on Zeno, that Zeno consulted the oracle (presumably Delphi) ‘to know what he should do to attain the best life’. The response of the god was that ‘he should take on the complexion of the dead’. Diogenes Laertius takes from his sources that Zeno’s interpretation of this is that ‘he should study ancient authors’. We can see that the true meaning of the oracle was both much more straightforward and much more profound than that.

6 Pythagorean Thought in Italy

Pythagoras was creating an eclectic doctrine, by syncretising elements from different sources. This is what Heraclitus means by saying that the collections of information in the three books are ‘poorly worked’. Not much interest has been shown in what Pythagoras, a long-time resident in Italy, might have drawn from Italian sources. In fact, it would seem that much of what was later passed off as Pythagorean in origin, actually has its origin among the Latins.

For example, the Romans also had a tradition of veneration of the boundaries and limits of things. Oskar Seyffert says of the god Janus that “even the ancients were by no means clear as to his special significance; he was, however, regarded as one of the oldest, holiest, and most exalted of gods”.

Of course, if the special significance of Janus was close to the heart of Roman religion, an absence of discussion might, rather than signifying a lack of clarity about his special significance, mean quite the opposite, and that the written tradition is quite misleading as to the Roman understanding of Janus, at least within the priestly community. 

“In Rome the king, and in later times the rex sacrōrum, sacrificed to him. At every sacrifice he was remembered first; in every prayer he was the first invoked, being mentioned even before Jupiter”. Which is indication of high status. If we recall the remarks of Pythagoras on what comes first and why, we can see that the significance of Janus is extremely important indeed. This is further emphasised by the fact that “in the songs of the Salii (‘jumpers’ or dancers) he was called the good creator, and the god of gods; he is elsewhere named the oldest of the gods and the beginning of all things.” The Salii were an old Italian college of priests of Mars, said to have been originally introduced at Rome by Numa Pompilius, the legendary 2nd king of Rome. He was said to be a native of Cures in the Sabine country, and was elected king a year after the death of Romulus.

William Smith says that Numa Pompilius “was renowned for his wisdom and piety; and it was generally believed that he derived his knowledge from Pythagoras”. Given that the foundation of Rome is traditionally 753 BCE, this is impossible, since Numa and Pythagoras would have been two centuries apart. However, the fact that later the institutions of Numa were associated with Pythagorean influence suggests that there was a perception of a relationship between the doctrines of Pythagoras and the foundation of Roman religion. Smith continues: “…he devoted his chief care to the establishment of religion among his rude subjects”, and to giving them appropriate forms of worship. He was instructed by the Camena Egeria (Aegeria), one of the twelve nymphs in Roman mythology. Numa later dedicated the grove in which he had his interviews with the goddess, in which a well gushed forth from a dark recess, to the Camenae.

Seyffert continues regarding Janus: “It would appear that originally he was a god of the light and of the sun, who opened the gates of heaven on going forth in the morning and closed them on returning at evening”. Rather, Janus, being the divinity associated with boundaries, is associated with gates, crossings, risings and settings, beginnings and endings, and the daily movement of the sun is the most important visible instance of beginnings and endings. In course of time (Seyffert suggests) “he became the god of all going out and coming in, to whom all places of entrance and passage, all doors and gates were holy” [my italics]. He continues:

In Rome all doors and covered passages were suggestive of his name. The former were called ianuae; over the latter, the arches which spanned the streets were called iani.

Many of these were expressly dedicated to him, especially those “which were situated in markets and frequented streets, or at crossroads”. In the case of crossroads, Seyffert tells us that “they were adorned with his image, and the double arch became a temple with two doors, or the two double arches a temple with four”. The way Janus was generally represented was “as a porter with a staff and a key in his hands, and with two bearded faces placed back to back and looking in opposite directions.”

Further, he is also the god of entrance into a new division of time, and was therefore saluted every morning as the god of the breaking day (pater matutinus); the beginnings of all the months (the calends) were sacred to him, as well as to Juno; and, among the months, the first of the natural year, which derived from him, Ianuarius. For sacrifices on the calends twelve altars were dedicated to him; his chief festival, however, was the 1st of January, especially as in B.C. 153 this was made the official beginning of the new year. On this day he was invoked as the god of good beginnings, and was honoured with cakes of meal called ianuae; every disturbance, every quarrel, was carefully avoided, and no more work was done than necessary to make a lucky beginning of the daily business of the year; mutual good wishes were exchanged, and people made presents of sweets to one another as a good omen that the new year might bring nothing but that which was sweet and pleasant in its train.

For the Romans, this juncture of the year, like every other juncture over which Janus presided, was a region in which change was more possible, more likely, than at any other time. Therefore, any immoderate behaviour, any departure from the normal daily pattern of life, whether through a quarrel or some other unpleasantness, might easily have taken root, and they might have found their whole lives dislocated as a result.

Seyffert continues that:

the origin of all organic life, and especially all human life, was referred to him; he was therefore called consivius (‘sower’). From him sprang all wells, rivers, and streams; in this relation he was called the spouse of Juturna, the goddess of springs, and father of Fontus, the god of fountains.

7 The existence of Irrational numbers

It is generally supposed that the Pythagoreans understood the world to be rational in nature, and it had long been argued that rational numbers were the product of ratios of other numbers. Their belief in rational whole numbers seems to have been a principal concern, possibly because whole numbers are often commensurable. The ancient assumption that the world was a rational creation, was maintained at least at the level of open public discussion.

There is however a famous story about the discovery of irrational numbers by the Pythagoreans, and their utter horror at the discovery. The discoverer of irrational numbers was supposedly drowned at sea, perhaps in consequence of this discovery. In fact, the story is likely to have a quite different meaning at an esoteric level, which I will discuss at the close of this paper.

So how was the Pythagorean proof of the existence of irrational numbers achieved? We should remember that The Eleatic school (home to Parmenides and Zeno, the former of which argued for the One and unmoving reality transcending the world of appearances) attacked Pythagorean doctrine by assuming their opponents' tenets, using the reductio ad absurdum technique to examine their credibility. The effect of such arguments was to reinforce the importance of the incommensurate in the world of number.

The Greeks attempted to extricate themselves from these difficulties by distinguishing between things which they would have preferred to have been commensurable (numbers and magnitudes), thereby rendering them incomparable. So the diagonal of a square could be regarded as a magnitude rather than as a length equal to the ratio of two numbers. By this means, irrational numbers could be largely ignored (a similar convenient fiction to one devised by Aristotle in connection with infinity, in which he subverted the difficulty of the infinite by dividing it into two: a potential infinite, and the actual infinite, which could be ignored).

From Thomas Heath:

We mentioned... the dictum of Proclus... that Pythagoras discovered the theory or study of irrationals. This subject was regarded by the Greeks as belonging to geometry rather than arithmetic. The irrationals in Euclid, Book X, are straight lines or areas, and Proclus mentions as special topics in geometry matters relating (1) to positions (for numbers have no position) (2) to contacts (for tangency is between continuous things), and (3) to irrational straight lines (for where there is division ad infinitum, there also is the irrational).

...it is certain that the incommensurability of the diagonal of a square with its side, that is, the irrationality of root 2, was discovered in the school of Pythagoras... the traditional proof of the fact depends on the elementary theory of numbers, and... the Pythagoreans invented a method of obtaining an infinite series of arithmetical ratios approaching more and more closely to the value of root 2.

Thomas Heath was writing at a time (1921) when classicists had very little knowledge of what was coming out of the ground in Mesopotamia and elsewhere, so his certainty that the school of Pythagoras ‘discovered’ the incommensurability of root 2 is a product of that time. He writes:

The actual method by which the Pythagoreans proved the fact that root 2 is incommensurable with 1 was doubtless that indicated by Aristotle, a reductio ad absurdum showing that, if the diagonal of a square is commensurable with its side, it will follow that the same number is both odd and even. This is evidently the proof interpolated in the texts of Euclid as X. 117... [Heath, T. (1921) Vol. 1 pp. 90-91].

It is a proof based on the law of non-contradiction. However, it is the consequence of the properties of Pythagorean triangles as they are represented to our understanding. The point of this demonstration is that: how things are represented to us is not the same as how they actually are. Or how they are in what we might term ‘transcendent space’.

Heath continues:

We have first the passage of the Theaetetus recording that Theodoras proved the incommensurability of root 3, root 5…. Root 17, after which Theaetetus generalized the theory of such 'roots.'... The subject of incommensurables comes up again in the Laws, where Plato inveighs against the ignorance prevailing among the Greeks of his time of the fact that lengths, breadths, and depths may be incommensurable as well as commensurable with one another, and appears to imply that he himself had not learnt the fact till late, so that he was ashamed for himself as well as for his countrymen in general.

This is interpretation about what Plato is saying which isn’t warranted. Plato was quite plain elsewhere (in the Republic and the Timaeus) that all things may pass into one another, and hence are in some way commensurate. He says this in connection with the Forms. As a general statement, it implies that the same is true for both commensurable and incommensurable numbers. We find ourselves in a strange place where the incommensurate may also be commensurate. Heath continues:

But the irrationals known to Plato included more than mere 'surds' or the sides of non-squares; in one place he says that, just as an even number may be the sum of either two odd or two even numbers, the sum of two irrationals may be either rational or irrational. An obvious illustration of the former case is afforded by a rational straight line divided 'in extreme and mean ratio' (Euclid XIII. 6) proves that each of the segments is a particular kind of irrational straight line called by him in Book X an apotome; and to suppose that the irrationality of the two segments was already known to Plato is natural enough if we are correct in supposing that 'the theorems which' (in the words of Proclus) 'Plato originated regarding the section' were theorems about what came to be called the 'golden section', namely the division of a straight line in extreme and mean ratio as in Euclid. II. 11, and VI. 30. The appearance of the latter problem in Book II, the content of which is probably all Pythagorean, suggests that the incommensurability of the segments with the whole line was discovered before Plato's time, if not as early as the irrationality of root 2. [Heath, T. (1921) Vol. 1 pp. 304-305].

8 Religious aspects of Pythagoreanism

Pythagorean thought is therefore a species of transcendentalism. It is a pattern of thought which understands reality itself (whatever that may be) as a principal concern, and as something which, as it is, transcends mundane earthly reality.

Within this pattern of thought however, earthly reality has properties and characteristics which have counterparts in the divine world. If 'God is Great’ for example, there are earthly examples of greatness, and so greatness is understood to be a property held in common between the worlds. What is held in common was understood by those of a transcendentalist persuasion to offer a connection between the worlds.

In essence the transcendentalist outlook among the Greeks holds that Being, or the ultimate reality, is both transcendent, and also present in the physical world. It is hard to imagine how such a view could arise except as the result of sophisticated logical discussion of the nature of reality. The idea defies common sense, and is counter intuitive.

This view of the world represents a paradoxical understanding of reality, in that the divine both transcends mundane reality, but is also at the same time present in every aspect of that reality. The connections between reality itself and earthly reality are not obvious, and often not easy to discover. The difficulty in discovering the connections is an index of the distance between the worlds. Yet it is possible to discover these connections. Reading the mind of the divine was of course a major concern in antiquity, since interpreting divine intention conferred knowledge and earthly power.

Working on or with the gods (theurgy) is often thought of these days as some obscure form of theological lunacy practised by the Neoplatonists and a few other groups in the dying days of the Roman empire. It is however a very old idea, based on the understanding that the sacred and profane worlds are connected with each other. It is also built into Plato’s account of the creation of the cosmos (in the Timaeus). The practice of theurgy is a corollary of the transcendentalist outlook, since if reality is transcendent, but we are also paradoxically part of it, then human will and intention are important to the way in which the world works. Physical reality does not represent a copy of transcendent reality, which Plato labelled as a likelihood only, but rather a subjective understanding of that reality *[note 1].

According to this way of thinking, there are processes which we can use to enhance what we have in common with divinity. Most often this was expressed in terms of gaining knowledge of the divine, since the supreme divinity was necessarily the fount of all knowledge (both Plato and the Mesopotamians concur on this point). If theurgy is an important component in early religious practice, it tells us something about how much knowledge was prized at the time, and also something of the scope of that knowledge in antiquity. Unlike homeopathic magic, which theurgy sometimes resembles, the practice of theurgy is entirely dependent on an understanding of Being for its effective use.

9 The Pattern of Eternity

That Socrates says at Phaedrus 247c that he dares to speak the truth concerning the nature of the region above heaven implies strongly that it is dangerous to do so - and after all, one of the charges against him was that he made theological innovations. Xenophon suggests that, though he was not formally charged with disbelief in the gods per se, Socrates was suspected of a form of atheism ["And how could he, who trusted the Gods, think that there were no Gods?" Memorabilia Bk 1 ch.1.5]. To hold the ultimate reality to be virtually indistinguishable at root from chaos, a place devoid of justice, beauty, order, etc., (and without location in time or space) except in potential, would be indistinguishable to the ordinary citizen from atheism. No wonder therefore that Plato writes the ironical words at Tim 40d:

Concerning the other divinities, to discover and declare their origin is too great a task for us, and we must trust to those who have declared it aforetime, they being, as they affirmed, descendants of gods and knowing well, no doubt, their own forefathers.

And at Tim 29a, concerning the model after which the universe was patterned, Timaeus asks:

Was it after that which is self-identical and uniform, or after that which had come into existence?

The latter implies change and disorder; therefore

if so be that this Cosmos is beautiful and its Constructor good, it is plain that he fixed his gaze on the Eternal, but if otherwise (which is an impious supposition), his gaze was on that which has come into existence.

Which is no more than an appeal to common sense. The nature of the arguments which might be adduced in antiquity to explain the world of appearance are, as the Sophist shows, much more complex.

Conditioned therefore both by the difficulty of the subject matter, and the social impracticability of the doctrine, we are forced to work out the doctrine for ourselves. That the method employed to convey the doctrine sometimes created unnecessary difficulties for the understanding, quite apart from its inherent difficulty is shown by the remark at 48c where some matters are not explained:

solely for this reason, that it is difficult for us to explain our views while keeping to our present method of exposition.

Nevertheless, the description of the Receptacle at Tim 50-51 is possibly the clearest exposition in Plato of the Real:

... it is right that the substance which is to receive within itself all the kinds should be void of all forms... that the substance which is to be fitted to receive frequently over its whole extent the copies of all things intelligible and eternal should itself, of its own nature, be void of all the forms... a Kind invisible and unshaped, all--receptive, and in some most perplexing and most baffling way partaking of the intelligible...

If then, Plato's unwritten doctrine (agrapha) placed chaos at the heart of Being, his conclusion would not be out of place among Greek speculations in general as to the nature of the arche: the difference is simply that he underpinned this conclusion with philosophical argument [Compare for example lines 116-128 of Hesiod's Theogony]. These we do not have for the earlier speculations, and therefore it is easy to conclude that they did not in fact exist; that the early speculations were not supported by cogent argument, and that the idea of chaos as the root or beginning of things never was any more than a concrete image of disorder. But Plato himself, putting the argument into mouth the of Timaeus at Tim 30a, uses such a concrete image, saying that God

took over all that was visible, seeing that it was not in a state of rest but in a state of discordant and disorderly motion, He brought it into order out of disorder...

We have virtually the whole of the Platonic corpus: of the earlier philosophers we have fragments like the one above. We should be cautious in presuming the absence of clear reasoning behind images simply because we have no direct access to such reasoning: that we do read concrete conceptions into the concrete images of the Presocratics is partly due to the fact that this was often the practice among the ancients themselves, and partly because, building upon this fragmentary and distorted evidence, we can frame a satisfying scheme in which there is a beginning, middle and notional end to the history of ideas, starting with concrete images and working up to pure abstraction.

10 Pythagorean Syncretism

It is important to recognise that the syncretism of Pythagoras draws on mathematical and geometrical ideas, as well as religious ideas. We normally choose to keep these separate. We imagine that they are separate. However, it is clear from the discussion of Pythagorean mathematics, number and geometry, that they perceived the necessary impact of the various puzzles and paradoxes which investigation of mathematics and geometry had on their view of reality. These were not parlour games.

Pythagoras was putting together a new religion, rather than a secular philosophy. It is unlikely to have occurred to him that a secular philosophy was possible, or for him to imagine what that would mean. We think of Pythagoras as a philosopher, because of how we understand what came after Pythagoras and his school. It is possible for us to so distinguish religion and philosophy, because we have lost sight of some very important aspects of how the gods were understood in antiquity. Pythagoras was well aware of the importance of the mathematical and geometrical aspects of religion, which is why he included them with the materials that we more naturally understand as religious ideas.

We know that Pythagoras drew on many sources for what became known as Pythagoreanism. He is likely to have drawn on both Italian and Greek ideas, and, as already mentioned, he travelled in Egypt, talking with the priests of the various cults (though we are told that most of them were not much interested in answering his questions); and also in the Levant, Arabia, and Babylon. He borrowed from them too.

We are accustomed to thinking that the intellectual life of these disparate cultures must have been as distinct as their iconography, their mythologies, languages, and systems of writing. But it is not necessarily so. Much of what we think we understand of ancient religion is the product of a more or less modern view, which sees a continuity between the religion of the common era and antiquity. So, since ’rational belief’ concerning the divine, rather than actual knowledge of the divine was (and is) of great importance in the major religions of the common era, it is assumed that ancient religions drew their strength from the same source, and are qualitatively similar phenomena. Modern scholarship is able to hold this view because, since the Enlightenment, we see the phenomenon of religion as irrational. The behaviour which supported ancient cultic life (sacrifice, divination by entrails, the worship of statues, etc.) is clearly more irrational than medieval religious practice, so there is little about it which demands the application of modern critical thought.

If belief is what is important in ancient religion, then we have missed nothing. If however there is a technical substrate to ancient religious thought, a substructure which depends on a combination of logical analysis, number theory, mathematics and geometry, then we have missed almost everything. Such a substructure does exist, and Pythagoras was aware of it, which is why religious precepts, number theory, mathematics and geometry were all present in his work.

It is possible to make a list of things which are part of this technical substructure in the religions of the ancient world. These are:

Extremity, the Mean, Totality, Perfection, Completion, Invariance, Integral (whole) numbers, the Incorruptible, the Commensurate, Greatness, Rising, Setting, Beginning, Ending, Duration, Periodicity, Points of transformation. And so on.

This list illustrates some of the things have exemplars on both the earth, and in the sky. These characteristics would, within this conceptual model of Pythagoras, have been understood to provide points of contact, and a bridge to the divine.

Why would Pythagoras want to create a synthesis of key components of ancient religions? There are many possible reasons, but the most important may be the intention to restore the technical level of religious thought and practice, then experiencing a long slow decline, so that number, mathematics and geometry might serve again, to make sense of the transcendental understanding of reality.

11 Pythagoreanism and the Deep

Returning to the question of the Pythagorean disciple who drowned at sea, we are told that the drowning occurred because, either he had discovered irrational numbers, or because he had divulged the fact that they exist (the sources for the story are inconsistent, which is often a pointer to glossed interpretation). In the 1st and 2nd millennia B.C.E, 'Ocean' was an image which referenced the idea of Being. Like Being itself, ocean seems without limit, and to be without form, shape and colour. It was an idea which was common to the Greeks and to Near Eastern cultures.

In Mesopotamia, there was an important story which told how man was first educated in the sciences, agriculture, and land-measure, by an amphibious creature (the sage Oannes) who emerged from the sea in the daytime and conversed with men, before disappearing back into the deep in the evening. As a creature of the ocean, and a sage of Being itself, he had access to all knowledge.

The idea of this is reduplicated in the more famous Mesopotamian story of Gilgamesh, which opens with the protagonist diving down to the depths of the sea. This makes sense once it is understood that the poem was known to the Mesopotamians as: ‘He who saw the deep’, meaning that Gilgamesh had access to knowledge of divine things. Perhaps the real meaning of the story of the drowning of the Pythagorean disciple is that, in understanding the fact that there are such things as irrational numbers, and that both irrational and rational numbers can be commensurate with each other, he was in possession of an esoteric understanding of the divine, which lay at the heart of the unwritten doctrine of the Pythagoreans.

Notes

1. In William Sullivan’s The Secret of the Incas, it is argued that the Incas were attempting to turn back the precession of the equinoxes, in order to preserve a heavenly bridge that they imagined gave them access to the divine world. The subtitle of the book is: myth, astronomy and the war against time. They came to the view that they could turn back time because of a transcendentalist understanding of reality, and the place of the Incas within it. It is a completely counter-intuitive outlook.

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