Thursday, 22 March 2018

Chapter Descriptions for 'The Frankish Tower' (forthcoming, 2018)

The chapters are summarised below.

Bertrand Russell and Philosophy in the West

Russell discussed the philosophical outlook of George Berkeley, the late 17th/early 18th century Divine, in his History of Western Philosophy, as it emerges from a conversation between two characters in a dialogue, Hylas and Philonous. This comes to a rather eastern flavoured conclusion, that the reality of the world involves cosmic mind, and that its physical nature is of very little importance. Bertrand Russell summarised part of this dialogue, but lost interest in the argument when it became an argument about theology, and about the divine. He was after all writing a history of western philosophy, and not a history of all varieties of philosophy. But I’m interested in the relationship between philosophy and theology in the west, and also the relationship between western philosophy and eastern philosophy. And Russell’s book won’t tell you much about either of those, because he wasn’t interested, and it wasn’t important. 

The Irrationality of Atheism

 Atheists do not deny the existence of the world, its laws and properties: they just argue that the concept of God is not required to accept the world, and to have an understanding of it. But this leaves them at a loss to explain how the world came to be, and why it should have come into existence.

Richard Dawkins and Deism

Modern atheism is actually dependent for its nature on the ontological argument, and the terms in which it is framed. Meaning that eight hundred years of argument about the nature and existence of God underpins the point of view of those who regard themselves as atheists. Dawkins makes a distinction in ‘The God Delusion’ between theism and deism. Theism is a pattern of belief which enshrines the idea that the Divine is responsive to man, and his rituals of worship and prayer. It is a pattern of belief dependent on the idea that God can act in the world.  By contrast, deism contemplates the idea that a creator God has existence, and necessarily created the world, but that he is not active in the physical world beyond that. This essay argues that Dawkins is in fact a modern deist rather than an atheist.

Contra Plantinga

Alvin Plantinga was kind enough to accept a copy of The Sacred History of Being. I sent two supplementary emails which outlined the implications of its criticism of the traditional ontological argument, whose function is to support a rational basis for belief, which are reproduced here.

Distinguishing Belief and Faith

Modern scholarship has a track record of making easy assumptions about the continuity of religious ideas and patterns of practice, and the accompanying social compacts. At the time the Assyrian palaces, temples and cities were being dug from the sand and soil in northern Mesopotamia, it was assumed that the relationship between the royal and temple establishments could be understood in terms of a modern division between church and state. This notion turned out to hold very little water on close analysis. It is also the case that belief is not a conspicuous feature of ancient religions.

The Death of Socrates

It is a puzzle that, in the midst of a thoroughly polytheistic culture in Athens, with its plethora of gods, its many cults and priesthoods in the service of those gods, that both Socrates and Plato could speak of ‘god’ in the singular. Our difficulty here is the result of a modern understanding of the significance of polytheism, which sees the phenomenon as the inevitable precursor to monotheistic belief, which excludes other gods from consideration, or credibility. For modern scholars, polytheistic belief in ancient Greece was something which developed, higgeldy-piggeldy, out of a plethora of local and tribal deities, much embellished with myths about their lives and actions, which served important social functions, but which had no universal meaning, and were not rooted in a model of reality which embraced consideration of what the nature of reality itself might be.

Logic, Sophistry, and the Esoteric in Ancient Education

Both Plato and Aristotle's writings contain arguments which either don't make clear logical sense within themselves, or in the context of the rest of the work. Sometimes the clues to the meaning of arguments are present elsewhere in the canons of both Plato and Aristotle, and some of them clearly involve an esoteric level of understanding. The whole body of their outputs need to be taken on board in order to grasp the meaning of individual works. This is usually not done with the works of Aristotle: his Historia Animalium is read by biologists and specialists in animal taxonomies, but usually they read little else of his work.

Beyond Mathematics and Geometry  

The process of separating ourselves from an interpretation of the world in terms of simple apprehension is driven initially by the practical necessities of our existence. But this process does not need to stop there. Intelligence consists in being able to adjust the categories of our understanding so that we do not mistake one thing for another. It is a mental development which might have no end. This is essentially how Kant understood human intellectual development, which he framed (in his Prolegomena) in terms of a general theory of a priori concepts, not based on empirical sense data, or even a mathematical or geometric understanding of anything in the world.

Obscured by Clouds: A Review of A.W. Moore’s ‘History of the Infinite’ 

This chapter is a critical response to Adrian W. Moore's radio series 'The History of the Infinite', broadcast in the autumn of 2016, and his book 'The Infinite', published in the early 90s. His treatment of the subject hardly references Plato at all. Adding Plato to the discussion changes the way in which the argument should be framed. Adding persuasive Mesopotamian evidence to the discussion shows how off the mark the argument is. The actual infinite is the principal source of ancient ideas concerning the divine, not Aristotle's potential infinite, so Moore's argument concerning our knowledge of God is forced to take refuge in the quasi-mystical Calvinistic idea of a 'sensus divinitatis'. His argument also makes it impossible to understand Kant's treatment of religion.

The Enlightenment of David Hume. 

 Though Hume's empirical approach was not wholly successful, some of his intuitions expanded our collective understanding of how we perceive reality – for example, his insight that we have no actual knowledge of the process of causation at all, and only a customary expectation of causal process, was a powerful one. We can describe causal processes, we can differentiate the nature of different causal processes, and we can formulate rules in connection with them, but we cannot know how causality itself operates, or even be sure that a perceived causal relation, often observed before, will obey the implicit rule the next time it is under scrutiny by us. However, it is no longer clear that Hume was exploring his mental processes and understanding entirely within the framework of western secular thought. This chapter is based on intriguing research by Alison Gopnik.

The Frankish Tower: Heinrich Schliemann and the Reconstruction of History

At the time when Schliemann was active, there was a strong desire among classicists to know whether or not there was any factual basis for the ‘heroic age’ portrayed by Homer in the Iliad and the Odyssey, and physical evidence for the reality of the war against Troy. Schliemann responded to that desire for evidence. He did find Mycenaean artifacts, and found a city showing a number of development phases in approximately the right place. Unfortunately the city he found is too small to have been the real Troy. And Mycenaean artifacts do not prove the reality of the Trojan war.

The Esoteric Conception of Divinity in the Ancient World. 

 In my twenties, I was struck by the strong interest the ancients had in the idea of limit – in art, architecture, philosophy, and ritual. This interest did not much seem to engage modern scholarly attention, with a couple of notable exceptions. Initially I had no idea at all what the significance of the idea of limit might be, and no idea where pursuing it would take me. Or that it would lead to a book it would take me four years to write, and which would reframe my understanding of human intellectual history in the process.

Ancient Conjectures, and Fictive Intellectual History

Plato argues that we should always look to the ‘one true thing’. J.G. Frazer however argued that questions concerning Being (‘the one true thing’) were entirely barren, since nothing could be predicated of Being. This of course is a spectacular instance of intellectual blindness, by which the richness of the intellectual matrix of ancient Greek thought was spirited into nothingness. In antiquity, nods were made toward the notion that the discipline of philosophy might not have been first developed in Greece, including (tellingly) at the beginning of Diogenes Laertius' Lives of the Philosophers.  Plato after all argued against the idea that philosophy was invented by the Greeks in the Protagoras, saying that it was of a great age – perhaps contemporary with the arrival of peoples from Egypt, who settled in the Peloponnese, and in also Crete.

What is Sacred, and what is Profane?  

Each of the divine names of Marduk, the head of the Mesopotamian pantheon,  has a description, and each of the lesser gods can be understood as abstractions of aspects of the rational creation. They represent excellences in the world. Marduk represents the sum total of these. This is the clue to understanding much of the ancient understanding what the divine is. Each described excellence resembles reality itself in terms of its properties. The excellence may serve social functions, as does a skill or specialism, but it should be performed for its own sake. The performance of these excellences recalls the perfection and completeness of the plenum, and reinforces the presence of the divine in the world. 

Polytheism, Monotheism, and the Cult of the Aten

The Aten is first mentioned (to our knowledge) in the Story of Sinuhe, which dates at least as far back as the twelfth dynasty, where the dead king is described as uniting with with the sun-disk in the heavens. Akhenaten’s iconography never shows the god in anthropomorphic form – instead the Aten is always shown as the sun disk with rays of light extending from it, with hands at the end of each ray. The Sun god was considered to be neither male nor female, but both simultaneously, an idea which was reflected in the depiction of Akhenaten in sculpture and reliefs. His full title however was ‘The Ra-Horus who rejoices in the horizon, in his/her Name of the Light which is seen in the sun disk’. We find this full rendering of the Aten’s name on the stelae placed around Akhetaten, which was Akhenaten’s newly founded capital. Sometimes the full name was shortened to Ra-Horus-Aten, or just ‘Aten’. Since two of the names of Akhenaten’s god refer to the sun (Ra being an older name for the sun god), it seems that some kind of intellectual synthesis of older ideas had taken place.

Stone Circles, Phenomenology, and the Neolithic Mind  

The evidence from the megaliths makes the importance of the sky very clear: in Britain and around the megalithic world, the sky was seen as a representation of divinity, of Being. As an image of the divine, it was an image of totality itself. The megalithic observatory, or temple, according to this hypothesis, was a device to embody aspects of divinity, of Being, actually in its structure, in the same way in which the gods in Mesopotamia might be invited to occupy their representations on earth.

Patterns of Thought in Late Neolithic and Early Bronze Age Britain

Pythagorean elements detected in megalith circles in ancient Britain have no easy explanation, and precede 1st millennium Pythagoreanism by an extraordinary period of time. This chapter 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 1st millennium, for which we have extensive documentation, It is suggested that this connection is a logical one. It is therefore possible that similar conclusions were arrived at in the Late Neolithic and in the Bronze Age.

Thomas Yaeger,
March 22, 2018

Thursday, 8 March 2018

The Frankish Tower (2018)

A collection of essays on philosophy, ancient cultic thought, and the problems of addressing and interpreting ancient evidence. Forthcoming, 2018.

The text is now complete for this, though there is some editorial and formatting work still to be done. The chapter list will be something like:



Bertrand Russell on Late Antiquity and the Middle Ages.
The Irrationality of Atheism.
Richard Dawkins and Deism.
Contra Plantinga. 
Distinguishing Belief and Faith.
The Death of Socrates.
Logic, Sophistry and the Esoteric in Ancient Education.
Beyond Mathematics and Geometry.
Obscured by Clouds: A Review of A. W. Moore’s ‘History of the Infinite’


The Enlightenment of David Hume.
The Frankish Tower: Schliemann and the Reconstruction of the Past.
The Esoteric Conception of Divinity in the Ancient World.
Ancient Conjectures, and Fictive Intellectual History.
What is Sacred, and what is Profane?
Polytheism, Monotheism, and the Cult of the Aten.
Stone Circles, Phenomenology, and the Neolithic Mind.
Patterns of Thought in Late Neolithic and Early Bronze Age Britain.

Select Bibliography.


The text will be around 50,000 words, and will be published by the Anshar Press. Precise publication details will appear on this page later in the year.

Monday, 5 March 2018

Patterns of thought in Late Neolithic and Early Bronze Age Britain

 Abstract: Pythagorean elements detected in megalith circles in ancient Britain have no easy explanation, and precede 1st millennium Pythagoreanism by an extraordinary period of time. 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 1st millennium, for which we have extensive documentation, It is suggested that this connection is a logical one. It is therefore possible that similar conclusions were arrived at in the Late Neolithic and in the Bronze Age.

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 BCE. 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 OT 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. 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.

It was once conjectured, on the basis of Alexander Thom’s surveys of megalithic circles, that there was a pythagorean element in these constructions in the late British Neolithic, and the early Bronze Age. This idea was later rejected (briefly discussed at the end of this paper). If the suggestion of a pythagorean element was in fact correct, that would push an extraordinary number of key ideas we associate with the 1st millennium BCE back into the Neolithic.

The level of engagement with Pythagoreanism which has been brought to bear on this question has so far not been significant. This paper is intended to provide a more sophisticated understanding of what Pythagoreanism implies, and how such an understanding can inform what sense we can make of such a very distant past.

2 Pythagoreanism in 1st 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, learning from the priests. On the other hand, we have information about the beliefs of the Gaulish priests from the mid-first century BCE, 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, 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 priestly 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 theology, cosmology, astronomy and natural philosophy.

Alexander Polyhistor 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 1st 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 In the Theaetetus. 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). is certain that the incommensurability of the diagonal of a square with its side, that is, the irrationality of {\displaystyle {\sqrt {2}}}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 {\displaystyle {\sqrt {2}}}roordroot 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 {\displaystyle {\sqrt {2}}}rorrrrrrrrkkk 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, {\displaystyle {\sqrt {3}},{\sqrt {5}}...{\sqrt {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 (Republic) 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 {\displaystyle {\sqrt {2}}}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 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 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 the three books of Pythagoras.

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. Transcendentalism in the Late Neolithic and early Bronze Age in Britain

Can we apply the content of this discussion to the Late Neolithic and the early Bronze age in Britain? If, for the purposes of argument, we make the assumption that just as Pythagorean number, mathematics, geometry, and the transcendentalist outlook were, in the mind of Pythagoras, necessarily connected with each other, these four things would also be present in megalithic culture in Britain and Gaul, for the reason that the missing piece in the record, the philosophical transcendentalism, is the necessary logical consequence of an understanding of number, mathematics, and geometry.

As we know from the studies made by Alexander Thom, the stone circles were built on the basis of various sizes of pythagorean right-angled triangles, and laid out with ropes of precise length. There has been some critical discussion of the ubiquity of the measure he described as the megalithic yard, which measured 2.72 feet, which he established by statistical analysis. However, if Thom identified different standard pythagorean triangles in the construction of different megalithic circles, all of which were based on the measure, then the presence and use of the measure is confirmed. It need not however, have been the only standard measure.

The construction process was designed and executed in such a way that the circumference of the circles, whether elliptical, egg-shaped, or flattened, would always be an integral number of the units used. This interest in integral numbers appears to have been universal among the builders of the circles.  The connectivity the integral numbers opened to transcendent Being is the reason why this was important.

This transcendent reality, understood to lie behind the physical world of appearances and its paradoxes (such as the essential identity of commensurate and incommensurate values), would be the principle focus of the megalith builders interest, and the design of the megalithic structures would have been understood to serve the function of strengthening the connections between the two worlds.  The transcendent world contains what is perfect, and the world of phenomena contains only approximations to such perfections. As Robin Heath pointed out in his account of Thom’s work, Cracking the Stone Age Code, the phenomenal world would have seemed to the megalith builders to be something of a crooked universe.

Looked at from this point of view, we can discern a significant motive in the geometrical construction of the major circles which Thom surveyed and analysed in detail. We can also begin to understand why there were different approaches to the construction of the circles, rather than a single standard design. In a crooked universe, there could be no universal answer to the problems they were trying to resolve. This universe is full of irrationality, simply because it is not the transcendent reality, but an imperfect representation of it. The irrationality could however be overcome in the physical world in specific instances of geometrical construction. In one case, by creating a design utilising an ellipse which measured precisely a specific multiple of the units employed in the pythagorean triangle used as the basis of the structure. In another, by making the structure egg-shaped, again with the same intention. The circle might also be flattened, in order to make the circumference commensurate with the units of the underlying triangle. 

But there is also the astronomical function of megalithic circles. As Thom identified, some are connected with the sun and its movements throughout the year. Others are keyed to the complex movements of the moon. For the later pythagoreans, and for Plato, the heavens represented a moving image of eternity. For these earlier pythagoreans, the heavens would be understood in the same way, and for the same reason.  A megalithic circle might therefore be conceived as a representation, in an abstracted form, of some the properties and attributes of Eternity. Eternity is something which is whole and complete, and returns into itself.

It therefore made sense to mark the extreme points of the movement of the heavenly bodies (which have their existence in the moving image of eternity), as a further embodiment of the connection between the worlds. These were constructed using only integral values, derived where possible, proportionately, from the movement of the heavens in relation to the earth. Heavenly cycles would be explored and represented in the structure where possible, together with indications of their periods. The motive for building the circles was performative, meaning that the structures served a set of religious functions on account of their existence and nature.

One of the objections made by the archaeologist Jaquetta Hawkes in the Chronicle documentary on Alexander Thom, made by the BBC in 1970, was that since the megalith builders did not have writing, there was no way of handing information on to succeeding generations. She also suggested that the inhabitants of the island during the period of megalithic culture were ‘simple farming communities... nomadic even’. But we know that the later pythagoreans cultivated memory. We are also told in Caesar’s account that becoming a priest in the late 1st millennium involved many years of study (around twenty), during which time a vast amount of information was committed to memory. So Hawkes suggestion that there was an absence of a means of handing on information is likely to be false. The cultivation of memory is built into the pythagorean view of reality, since what exists in the mind was understood to be more real than what could be understood by the senses.

12 Walking back the insight into Ancient Mind

Robin Heath has documented the archaeological community’s turning away from engagement with Thom’s work. This happened for a number of reasons. One of the reasons was that there was little that archaeologists could do with the information which he presented, and they had no idea at all what it might have meant. Thom surveyed and studied the stone circles, and inferred various properties, such as the apparent obsession with whole number in their construction, their use in eclipse prediction, and their connection with foresights in the surrounding landscape. Beyond this, Thom himself was largely unable to supply useful interpretative context to the phenomenon of megalith building.

Thom was also in conflict with a key assumption of archaeology, which is that man (in general) has been progressing, generation by generation, since the earliest times. The suggestion that there was a profound body of mathematical and astronomical knowledge so far back in time just didn’t fit with this paradigm. In the end it was argued that, despite the undoubted quality of his surveying of the monuments, he was seeing something that wasn’t actually there. 

A significant part of his work was resurveyed by the mathematician Clive Ruggles in order to determine the case. Ruggles’ approach involved avoiding any concern for the exact orientation and location of the foresights which Thom had identified, and so necessarily made that part of the evidence meaningless for the interpretation of the function of the sites. As a consequence of this approach, as far as the discipline of archaeology was concerned in the late eighties, there now was no longer a puzzle to be addressed.

The real problem for the interpretation of the stone circles however is the absence of any understanding in the modern world of a necessary connection between ideas of number, mathematics and geometry, and thought concerning the divine.  As suggested, we read ancient religions as analogues of modern religions, which we understand without reference to a technical substrate (though these substrates are sometimes still present in vestigial form).  For us, religion is about bodies of belief. So instead of an understanding of religion in terms of series of responses to fundamental philosophical questions, it is understood in terms of collective belief in socially useful behaviours, ritual, and myth. We understand ancient religion (as far as is possible) in terms of sociological, ideological, and sometimes pathological functions.

13 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 BCE, 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.


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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Chronicle (BBC), Cracking the Stone Age Code, 1970. The documentary film is available from the BBC Archive.
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There is a significant collection of papers and reviews concerning Thom's work available from The SAO/NASA Astrophysics Data System (ADS) which is a Digital Library portal for researchers in Astronomy and Physics, operated by the Smithsonian Astrophysical Observatory (SAO) under a NASA grant. It is possible to select the files you want, and to download them as a collection.

The BBC Chronicle episode from October 1970, Cracking the Stone Age Code, which is available from the BBC Archive. The file uses Flash. If it doesn't work first time, try reloading the page. It is also available on Youtube:

The archaeologist and anthropologist Euan MacKie gave a lecture (British Archaeology and Alexander Thom) to a lay audience in 2013, in which he discussed both Thom's work and the subsequent rejection of Thom's findings by the archaeological community. He also discusses the problematic nature of Clive Ruggles methodology in resurveying some of the megalithic monuments in Scotland which Thom examined. Essentially Ruggles methodology, conceived as a way of excluding subjective bias from the survey process, necessarily makes it virtually impossible to reproduce Thom's results.