Plato and the Transcendental Infinite

 

[This post is an extract from:'Evading the Infinite',   one of twenty-one essays in the book Man and the Divine, published in August 2018.  Information about Man and the Divine can be found here] Part of a critical commentary on Adrian Moore’s A History of the Infinite, broadcast in ten episodes by the BBC (on Radio 4) across two weeks in late September/early October 2016. The first episode was broadcast on the 19^th September. The book is available in ePub format from leading retailers of eBooks, such as Barnes & Noble, Blio, Kobo, Itunes, Inktera, Smashwords, etc.

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I have spent many years studying Greek philosophy, and as a result I found both Moore’s arguments and his narrative concerning the idea of the infinite to be oddly structured. There is a gaping hole at the start, since Plato is scarcely mentioned, and none of his arguments appear in the narrative (sometimes voiced in the dialogues by his master Socrates).  He does discuss the ideas of Pythagoras, but in such a way that it is hard to recognise him, and the many parallels which exist in Plato’s writing. As a result, this history of the infinite is not a complete history, tracing the discussion of the idea from the earliest period possible, but a history with a strong point of view, which begins at a point which is convenient for the arguments which follow (Moore’s book on the infinite has a much broader compass).

Part of my purpose here is to outline Plato’s engagement with the idea of the infinite, and to place it before Moore’s chosen point of departure. Understanding what Plato said concerning the unlimited and unbounded necessarily changes the interpretation of Aristotle’s views and arguments, with which Moore begins. Simply writing Plato out of the narrative not only creates something of a fictitious narrative, but also creates difficulties that otherwise would not exist.

Oddly for an account of man’s engagement with the infinite, the first of the series of programmes is titled ‘Horror of the Infinite’. Moore quotes the mathematician David Hilbert:

The infinite has always stirred the emotions of mankind more deeply than any other question; the infinite has stimulated and fertilized reason as few other ideas have; but also the infinite, more than other notion, is in need of clarification. 

Moore accepts Hilbert’s characterisation of the idea of the infinite. He begins by saying that

ever since people have been able to reflect, they’ve been captivated and puzzled by the infinite, in its many varied guises; by the endlessness of space and time; by the thought that between any two points in space, however close, there is always another; by the fact that numbers go on forever; and by the idea of an all-knowing, all powerful god. People have been by turns attracted, fascinated, perplexed, and disturbed, by these various different forms of infinity. 

Indeed yes. But Moore’s account appears to start at ‘disturbed’, rather than ‘attracted’.

Is God the Infinite, and Reality itself? Moore does not much concern himself with this question in this sequence of programmes, at least not in the terms in which the Greeks understood the question. The following is an extract from The Sacred History of Being (2015):

 The Greeks did not contemplate the idea that the ‘existence’ of God, or the supremely perfect Being, was subject to proof. This would have been anathema to them, for the reason that they understood the very concept of the divine is inevitably beyond the capacity of the human mind to understand, or to frame. It is also beyond space and time. It is possible to say something about the divine, but that is all. Saying that the supreme perfect Being has a property ‘perfection’ is fine, but the meaning of this perfection is strictly limited in its human understandability. To attribute the property of secular ‘existence’ to this Being would have been regarded as absurd.

Yet it would be granted that one could argue that, without the property of existence, the perfection, or the completeness of God, was compromised. But for it to be in the world of change and corruption would also be understood as compromising the perfection of the supreme Being. At least in terms of public discussion. Thus the Greek view of reality and the Divine was that there was a paradox at the root of reality and the gods, and that it was not possible to define the nature of the Divine without exposing that definition to contradiction. The enlightened enquirer into the nature of the divine therefore is spared further pointless argument about the nature and the very existence of God. Both are conceivably true. But the true nature of the Divine, being a paradox, rises beyond our capacity to argue about that nature. It remains a matter of conjecture.

Our human experience tells us we live in a world in which change is possible, and inevitable. The definition of the Divine on the other hand, tells us, the divine reality beyond this world of appearances is a place of eternal invariance. It suggests that at the apex of reality, it is not possible for the divine to act in any way, or to participate in the world of change. Again there is a difficulty if we hold that the greatest and most perfect Being can do nothing without contravening its essential nature. A whole range of properties would clearly be missing from the divine nature.

It would seem that the Greek solution to this problem was to argue, as Plato and the neoplatonists did, that the world of reality was in fact invariable, as the theory requires. And it did not at any time change. But a copy was made. As a copy it was less than perfect, and this imperfection created the possibility of change, action, and corruption. This copy is eternally partnered by the original, which stands behind it, unchanging and unchanged by anything which happens in the copy of the original divine model. As a copy it is the same, but as a copy it is different.

This however, is a solution which Plato labelled as a likelihood. Which is code for: ‘this is not the answer to the problem’. 

One of the properties of the supremely perfect Being would be that he was one and not two. In the creation of a copy, the invariability of the divine has been breached, and the divine is now two, not one. Two, not one, would seem to be a fatal objection. Firstly the copy is a representation of the original, and not the original itself. Secondly, the copy is imperfect, and through the act of representation, it has become different. The original continues complete in its original nature, with its original properties and characteristics.  Plato hints at territory beyond this contradiction, but does not venture into it overtly.

This is the key mystery of ancient thought. To understand the full significance of this problem, and its implications for ancient models of reality, we need to look closely, as they would, at what a copy of Being actually means. There can be no copy, at least not in an objective sense. And if there is no objective copy, then the world which moves and which has existence, must be a subjective view of Being.

Apart from anything else, if the world is a wholly subjective experience, occurring (if we dare to use that word) within Being itself, then the change and motion which is apparent to us, and which contradistinguishes the world of existence from Being, which is itself and only itself, must be illusory. The illusion may be convincing, but ultimately it remains as an illusion, however persuasive it is to us, that there is an objective reality which is subject to change and movement.

This is the correct answer to the problem. Our experience in the world is of finite things, which are finite representations of things which are infinite. But this world is also infinite, and at the same time. It is therefore a matter of apprehension, understanding, and will, if man is to engage with infinity, and reality itself.

Hence Plato’s discussion of the ascent to The Good via the Forms, to that infinite place where all knowledge is to be had, and to descend again with divine knowledge, again entirely via the Forms, to the world of sensibles. What he is actually talking about is a formal process and discipline by which the finite human mind can engage with infinity.

Pythagoras was much closer to Plato in terms of doctrine than scholars normally allow. I can demonstrate this by quoting the Neoplatonist Porphyry who wrote about Pythagoras many centuries after his lifetime. 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.

That is exactly the doctrine of the ascent and descent via the Forms which is described by Plato. 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’ 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.

What is transmitted to us via the writings of the Platonists, is something of the basis of both their understanding of what the Divine actually is (the Infinite, the Limitless, and Reality itself), and how man may have commerce with the Divine, through sacred rather than profane practices, in a world which has a double nature, and in which man has a choice.

Looked at in this way, rather than being a history of infinity, Moore’s argument is about the idea of the infinite from the point of view of finitude. This is the way Aristotle chose to deal with the infinite, by dividing the concept into the actual infinite, and a potential infinite, and dealing with the latter. Moore has said elsewhere that the way he treats the infinite is generally in terms of an Aristotelian Finitism.

We might pause here and consider what the implications might be of the identification of the Infinite and the Divine, which seems to be implicit in the views of a number of ancient philosophers. If they did so identify these concepts, then much of Greek religious thought and practice was based on a philosophical understanding of the infinite. In which case, Moore’s history is a history of what happens when the actual importance of the infinite in the life of man is forgotten, misunderstood, and eventually no longer noticed for what it is. Much of Moore’s argument is shaped by his Aristotelian Finitism.

In the first programme, Moore argues that the Pythagoreans thought finite things were good, and that infinite things were bad (this information comes to us via Aristotle), and that they thought they had evidence that the finite had some kind of control over what was infinite. And that the usefulness of rational numbers showed that this was the case. This is clearly a garbling of Pythagorean thought from a distant age, if Pythagoras thought that ascent to eternal and incorporeal things was important, as I’ve suggested. There is also discussion of musical ratios, and the Pythagorean discovery that different string lengths with simple ratios are more consonant to the ears than those which involve large values. Their ‘discovery’ of irrational numbers, which can be found using the theorem of Pythagoras, is said to have filled the Pythagoreans with horror, and the story of one of their number being drowned at sea after revealing their existence, is referenced. Rather than revealing their horror of irrational numbers, this is a story which points to their interest in whole numbers. The idea that they once had no idea about the existence of irrational numbers is absurd.  

The programme moves on to consider whether other ancient Greeks had the same resistance to the infinite. The views of Anaxagoras on infinite divisibility are discussed. Anaxagoras was relatively comfortable about these ideas. Zeno’s paradoxes in connection with infinite divisibility are also discussed, including his paradox of travelling by an infinite number of half distances, which seems to imply that movement is impossible. The similar paradox of Achilles and the Tortoise is also referenced. Observation and reflection thus seem to contradict each other. Zeno distrusted observation to the point that he believed that movement was impossible. Parmenides was Zeno’s teacher, and taught the universe to be a simple unity. So, only the appearance of motion is possible. Otherwise the universe would have to have infinite complexity. Moore winds up the episode by suggesting that because of these paradoxes, and the existence of irrational numbers, that there is some truth in the suggestion that the Greeks had a horror of the infinite.  

Looking at the content of this episode in the light of the added preamble about ideas of the infinite held by Plato and Pythagoras, we can see that something old and valuable is contained in the writings of some earlier philosophers, transformed into more or less secularised accounts of the arguments the Greeks used to illustrate the paradoxical nature of the infinite aspects of the world, as they manifest in the world of the finite. 

We  get many clues about the Greek understanding of the infinite and the unlimited from a number of Plato’s dialogues, including The Timaeus, The Sophist, The Republic, The Theaetetus, The Laws, and The Parmenides. In skipping Plato, the first reference to Parmenides and his notion of the universe as simply one and one alone, is as an introduction in the first episode to his pupil Zeno of Elea, and his response to paradox. There is no discussion of Plato’s demolition of Parmenides arguments, no discussion of the Platonic forms, no discussion of the relationship of the forms to the form of the Good, which is another way of talking about what is infinite, and no discussion of what amounts to a different logical modality in the pages of Plato (where he discusses things passing into one another by means of their similitude), which is a way of understanding the relationship of finite things to the infinite.  

Essentially Aristotle’s rapprochement, which Moore characterises as an attempt to make the concept of the infinite more palatable to the Greeks, involved dividing the idea of the infinite into two. As already mentioned, one of these was the potential infinite, and the second was the actual infinite. As outlined in the first episode, Zeno’s paradoxes depended on the idea of an infinite divisibility, which seemed to make the idea of any kind of movement impossible, since that would require a universe of infinite complexity. Zeno therefore regarded all forms of movement as illusion. Since in order to travel a certain distance, you would have to travel half the distance to your destination, and then half of the distance remaining, and then half of that, and half of what still remained, and so on. Which would result in an infinite number of steps. Which would be impossible. 

Aristotle’s response was that though the various stages of the journey could be understood in such a way, the stages were not marked, and did not have to be considered in making a journey. The idea of limit is however a crucial point. What Aristotle was saying is that there are two ways of looking at the idea of what a limit is.  Essentially there is limitation which is defined by what a thing is, and there is limitation which is not. In the first case the limit of a thing cannot be transcended without the nature of that thing turning into something else.

The essence of this argument is that there are forms of limit which can be ignored. One of which is the actual infinite: instead we should deal with the potential infinite. The actual infinite, by its nature, is always there. But we cannot deal with it. The potential infinite we can work with, since it is not always there, and spread infinitely through reality. So we can count numbers without ever arriving at infinity, or ever being in danger of arriving there. Moore mentioned that this conception of infinity more or less became an orthodoxy after Aristotle, though not everyone accepted that his argument against actual infinity was solid. Which is something of an understatement. Aristotle’s distinction between the potential infinite and the actual infinite is between what is, in practical terms, something we can treat as finite, and what is actually infinite. 

It might seem surprising that Moore’s first port of call in part three is the philosopher Plotinus, who was writing in the third century C.E., some five centuries after Aristotle. The reason that he has jumped to Plotinus is because he argues that Plotinus claimed not only that the divine was infinite, but that the divine was the infinite. Thus conflating the ideas of divinity and infinity in a way that – he says – no one had done before. Or, to be more precise, he declared the identity of the divine and the infinite in a way no-one had done before.  

Well no. As I’ve argued at the beginning of this essay, Plato’s principal interest was in a transcendent reality, which it would be hard to distinguish from the infinite, except in hair-splitting terms. He refers to the necessity of ‘looking to the one thing’, and that the ‘one thing’ is something which is found nowhere on earth. In one of his dialogues, he has Socrates describe that transcendent realm as something which possesses ‘no form, shape or colour.’ It is clearly without definition and limitation, with no finite properties and attributes, which means it is unlimited, and infinite. It is also the ultimate source of all knowledge. So it also seems to possess the properties and attributes which are associated with the divine. Plotinus’ supposed innovation is therefore no such thing. Anaximander’s understanding of the ‘apeiron’ (the unlimited) as the cause of all things is just such an equation of the divine with the infinite, which means the idea was around in the sixth century B.C.E. 

Adrian Moore on Georg Cantor and the Size of Infinity

 

The sixth episode of Adrian Moore’s radio ‘A History of the Infinite’ is concerned with the infinitely big, considered not in terms of physical size, but in the context of mathematics. It focuses on the work of the German mathematician, Georg Cantor, who devised a way of distinguishing between different infinite sizes, and of calculating with infinite numbers. Cantor was the first to do such a thing.  One of the most interesting developments in modern mathematics, and as Moore says, his work was ‘utterly revolutionary.’ 

Everyone knows there is no such thing as the biggest number. No matter how far you travel along a sequence of numbers, you can always count further. Even Aristotle, who Moore suggested in an earlier episode was an arch-sceptic about the infinitely big, accepted the reality of the infinite only in terms of processes and sequences which were destined to go on for ever. 

This might be a little tendentious, since as Moore has already pointed out in the episode ‘Aristotle’s Rapprochment’, he divided the concept of the infinite into two things: the actual infinite, and a potential infinite. The world of numbers and calculation exists in the context of the potential infinite, in which change happens in space and time. The actual infinite, for the purposes of mathematics, is simply ignored, since it is (apparently) not possible to work with it. I make this point since there is much about Aristotle’s wider philosophical work which points to a strong concern with the actual infinite. He isn’t sceptical about the reality of the infinite.

Aristotle’s view prevailed for over two thousand years, and during that period there was hostility to the idea that the infinite itself could be the subject of mathematical study in its own right. This orthodoxy was not challenged until the late nineteenth century, when Cantor presented a systematic, rigorous, formal theory of the infinite. Moore is interested in what drove him, and at what cost.

Cantor had a very hard time in trying to have his ideas accepted by the mathematical community, partly because of the perception that there was a religious component to his work. Henri Poincaré said of his work that: ‘it was a disease, and there would be a cure.’ His teacher Leopold Kronecker, who might have been expected to support his pupil, was hostile to his work, and made it difficult for him to publish. Kronecker said ‘God made the integers, all the rest is the work of man’. Cantor suffered several nervous breakdowns, possibly because of the sheer perplexity of his work, and died in an asylum.

Moore now considers set theory. How do you count without actually counting, and know if a set or collection is the same size as another? You can assemble pairs of things, such as male and female, cats, dogs, etc. If they are paired, and there are no extra males, females, cats or dogs left over, then you know that they are the same size without counting the individuals in the sets.

Does this apply to the infinite? Cantor asked why not? But here things get a little weird. The set of what Moore refers to as ‘the counting numbers’ (positive integers) appears to be the same size as the set of the even numbers. Even though the first set includes all the numbers in the set of even numbers, plus all the odd numbers. If we want to show the number of counting numbers is the same as the number of even numbers, we can do this fairly easily by pairing the counting numbers with the even numbers which result from doubling them. There will be nothing left over, so we can say that these two sets are the same size as each other. Moore says that it is tempting to say that comparisons of size just don’t make sense in the infinite case. But Cantor accepted that they were the same size, despite the fact that the first set contained everything in the second set, plus more besides. 

Can we use this technique to show that all infinite sets are the same size, which might not be a counter-intuitive conclusion? In fact, some infinite sets are bigger than others, as Cantor discovered. Even if you start with an infinite set, it will always have more subsets than it does have members. You cannot pair numbers with the subsets: there will always be a subset left over. So there are different infinite sizes. Moore does not draw the conclusion that it is the unbounded nature of the infinite which makes the differently sized infinities true. What is infinite contains all things which are possible. It is not just something which is extremely large.

Cantor’s work polarized opinion in his lifetime, and it has continued to polarize opinion ever since. The mathematician David Hilbert famously said ‘No one shall be able to drive us from the paradise which Cantor has created for us’. To which Wittgenstein responded: 'I wouldn’t dream of trying to drive anyone from this paradise: I would do something quite different – I would try to show you that it is not a paradise, so that you leave of your own accord’

Moore concludes with a question: “Is Cantor’s work of any significance outside mathematics? Some would say that it is not. It certainly made its mark by creating as many problems as it solved.”

It can however be argued that many difficult questions are difficult for us as the result of an important concept dropping out of western philosophy, which is the concept of the plenum. This concept is not discussed by Moore in this series of programmes. The idea of the plenum is that reality itself is undifferentiated possibility, something which does not exist in time and space, but contains every possible aspect of time and space, and everything which might be contained in it as potential, as something which might be generated within physical reality. With the idea of such a transcendent reality, almost anything which can be imagined to exist, can have existence. But such things will inevitably point back to the nature of the initial plenum in some way, and be full of puzzles and paradoxes. In rejecting this view of infinity, and treating it as if it had no bearing on sensible reality, Aristotle and those who followed afterwards, effectively closed off the possibility of understanding why such paradoxes exist in the physical universe.

In the seventh episode there is a brief introductory recap, reminding us that Georg Cantor created a formal theory of the infinite in the late nineteenth century. The impact of his work on mathematics was large, and led to a period of unprecedented crisis and uncertainty. Subjecting the infinite to formal scrutiny, led to mathematicians confronting puzzles at the heart of their discipline. These puzzles indicate some basic limits to human knowledge.

Moore invites us to consider the issue of sets of sets. How can there be more sets of sets, than there are sets? He suggests at this point that our heads may begin to reel. But why shouldn’t we have, say the set of sets which have seven members? Enter Bertrand Russell, who, in trying to come to terms with some of these issues, arrived at what is known as Russell’s Paradox. He argued that once we have accepted that there are sets of sets, we can acknowledge sets which belong to themselves, and those which don’t. A set of apples is not a member of itself, for example, since it is not an apple.

The paradox arises in connection with the set of all sets which are not members of themselves. On the face of it, there should be such a set, but there is not. For the same reason that there cannot be a nun in a convent who prays for all those nuns in that convent who do not pray for themselves. This is a matter of logical rules. She is going to pray for herself, only if she does not pray for herself, which is impossible. Russell’s paradox seemed to indicate a crisis at the heart of mathematics, where sets play a pivotal role.

 Russell communicated his paradox to the German mathematician Gottlob Frege, which is a well-rehearsed incident in the history of philosophy and mathematics. Frege had been trying to put these mathematical issues on a sound footing in a three-volume work, which was two thirds completed. Russell’s paradox came like a bolt from the blue. Frege replied saying he was ‘thunderstruck’, since the paradox undermined his attempt to give a sure foundation to arithmetic, while he was engaged in writing and publishing his life’s work. Frege died embittered. 

Returning to Cantor, Moore discusses his work with the problem of the ‘counting numbers’, (1,2,3,4, etc), which constitutes a smaller group than the group of possible sets of the counting numbers. The question arose of how much smaller the first group was. Cantor’s hypothesis was that it was just one size smaller, and that there were no sets of intermediate size. But he was unable to confirm that this was the case, or to refute the idea. So he was in a state of uncertainty for a long time, and this exascerbated his lifelong problem with depression. This question was listed by David Hilbert as one of the 23 most important questions in mathematics to be addressed in the ensuing century. 

The matter is not settled, even now. Is this the result of mathematicians not being assiduous enough? Moore says that it has been shown that it is impossible, using all of the tools available to mathematicians, to resolve the issue. It looks as though we are stuck with an unanswerable question.  Perhaps not completely unanswerable, but it is with the toolkit of mathematical principles which are currently available. No new principle has been discovered in the decades since, so it looks as though we have stumbled on an inherent limitation on mathematical knowledge. 

The logician Kurt Gōdel showed that this limitation was in a sense unavoidable, in that, with a limited set of mathematical principles, there will always be truths which lie beyond their reach.

So there are many questions about the foundations of mathematics, and their security, or insecurity. Russell’s paradox of the set of all sets which don’t contain themselves, had revealed an inconsistency in the principles mathematician’s had been working with up to then. David Hilbert had said “how do we know there isn’t another inconsistency elsewhere in mathematics generating the problem?” He devised a programme to map mathematics with a limited but very precise set of principles, in order to discover if this was the case. Gōdel’s work however, made it unlikely that this programme would be a success. Is there a crisis in modern mathematics? It was suggested that modified versions of the Hilbert programme have proved that there are no other inconsistencies in basic mathematical principles. And that consequently the rest of mathematics is essentially reliable and consistent. Moore concludes that mathematical work on the infinite has left us acutely aware of what we do not know, and indeed what we cannot know

 

The Paradoxical Nature of Reality

 

 

[This is the postscript to The Sacred History of Being, first published in November 2015. It makes the case that we live in a reality which has two states, and that moving between these states has been a major preoccupation of ancient civilizations. The statue is of Amenhotep III, who was the father of Amenhotep IV,  who later changed his name to Akhenaten. Amenhotep III is shown wearing the Khepresh Crown, also known as the Blue Crown. "During the New Kingdom, pharaohs were shown with this crown in military circumstances. However, some scholars think that the crown was also meant to evoke the divine power of the pharaoh, and was thereby worn to religiously situate kings as manifestations of gods on earth."] 

 

The argument of this book is complex and necessarily discursive, since its purpose is to uncover an ancient implex of ideas which was almost always discursively expressed, whether in the course of argument, in the formation of myth, liturgy, or ritual, and was, for the most part, conjured in terms of images. The argument however has a clear focus, which is the consistent reference point of these images - that which does not change - and secondarily the body of ideas which grew up around this notion of a reality which is beyond all merely human understanding.

 

We know that this idea was pursued in Greece, in particular by Plato, who was 'always looking to the one thing.' This 'one thing' has been very difficult for students of Plato to disinter, since it seemed to be hedged about with logical difficulties which, not least, cast considerable doubt on the possibility of there ever being something which could be referred to as the 'one thing.'

 

This difficulty isn't simply the result of the coy and often allusive way in which Plato discusses the relationship between the world of change and the world of unchanging reality. It also has something to do with us, and the weight of cultural baggage which we bring to bear on certain questions. We read Plato with many assumptions.

 

It is also the case that we do not read Plato (or any Greek philosophical writer for that matter) in cultural context. The Platonic canon is about philosophy, and philosophy is currently treated, almost universally, as a subject which can be abstracted from its original context without any significant damage to its meaning or worth. What that original cultural context is, is difficult to determine.

I have attempted to turn this assumption on its head. The Platonic canon belongs very precisely in its cultural context, and we do damage to our capacity to understand both Plato, and the culture to which the canon belongs, by breaking these connections without any grasp of what is lost.

 

The culture of Greece in the 5th and 4th centuries B.C.E is generally interpreted by classicists in terms of something which happened in isolation from other cultures around the Mediterranean, because it is assumed that the culture of Greece is, in essence, an autocthonous development. The loss this creates for our understanding of all of the cultures of the first millennium B.C.E - including Greece itself - is colossal. Greek intellectual life is relatively well documented when compared with other cultures around the Mediterranean; but rather than using one to illuminate the other, the classicist and philosophical communities assume that they are so far apart, no meaningful comparisons can be made. 

 

 I have attempted to stand this assumption on its head too, by comparison with aspects of the culture of the Neo-Assyrian empire. Some key details of Greek thought and religious life, which are entirely missing from the record, are documented in Assyria (and in Babylonia) in the mid-1st millennium B.C.E. In writing about the Forms, Plato is, as a key passage indicates to us, also writing about divine statuary and their cultic function. The Forms, as discussed in the canon, reflect a hierarchy which leads to the contemplation of the 'one thing.' This one thing Plato referred to as 'The Good'. This is the ground of Being, or reality itself. All of the Forms participate in some way in Being, and represent aspects of it. Being is conceived by Plato to be the home of real knowledge, and this knowledge can be accessed via the hierarchy of Forms, contemplated in the mind. The Forms represent perfect or ideal approximations to the perfection of Being itself, and provide the access route.

 

Assyria too possessed the ultimate abstract idea of Being also, some considerable time before Plato wrote about the idea. We know this because of the close relationship which has been established between the structure of the Kabbalah and the Assyrian sacred tree, which is represented many times in royal palaces and on seals. The Assyrians regarded the pursuit of knowledge as an essential component of kingship, which we know about from several sources, including a personal account of the scribal training of Ashurbanipal. The wider training of the king involved the pursuit of excellence in various skills. The king was understood to represent divine Being on earth.

 

'Looking to one thing' was also essential to the Assyrians, since they too conceived that knowledge resided in transcendent Being, which, they called by many different names. One of the principle images of Being is the undersea Abzu, the home of Ea, the god of wisdom. This is the place where generation is made possible, and where the destinies are determined. Assyrian rituals for inaugurating divine statues invoke the beginning of creation in the Abzu, and feature a long series of images which repeatedly point toward this idea of Being. And in order for a statue to become a living being in heaven, it must be (and is) granted the wide knowledge of Ea himself.

 

Plato's account of the creation involves the use of an image of Being as a pattern for the generation of the world. This image of Being was used by the Demiourgos to pattern the heavens, with its constellations and moving planets. It is therefore striking that the making of a divine statue in Assyria involves a ritual which exposes the statue to the heavens in order that it may have knowledge of the divine. The parallel Babylonian ritual goes into more detail about the aspects of the heavens which impart divine knowledge to the new god.

 

In both Assyria and Greece generation is associated with the sea. In both cases the sea is understood as an image of the limit of reality - Okeanos, the encircling ocean in the case of Greece, and the upper and lower sea, in the case of Assyria. It is the limit which is significant, and which gives rise to the image of it. Plato has much to say about the transcendent realm of The Good, in which Forms can have no location, size, shape or colour. The limit appears as the site of generation in Assyrian ritual in the form of the riverbank, which was regarded as the gateway to the Abzu and its threshold. Similarly therefore, in Assyria, generation is understood to emerge from a place which has no location, size, shape or colour. The pure reeds of the riverbank are stated to have their roots in the Abzu.

 

The precise nature of the cultural relationship between the cult of knowledge in Greece and in Assyria, is currently unclear, apart from the obvious parallels. It will take time and effort to clarify this. But we do know that another important philosophical idea was held in common both in Greece and in Mesopotamia, and we know the nature of the doctrine which emerges from it.  This is the doctrine of wholes or totalities, which we are told was acquired by Pythagoras while he was in Babylon. This doctrine concerns how ideas and forms can participate in one another. It seems like a small thing, outside its proper context. But the doctrine has huge importance in understanding ancient philosophical theology, and indeed in knowing that it existed at all. Anything which is whole, or total in some respect, was understood to participate in wholeness or totality itself. Being was understood in both cultures as a form of totality. In fact in both cultures, it was understood as the entirely transcendent form of totality.

 

Plato employs this doctrine in his account of the creation, and it is the means by which it is possible for one thing to pass into another, as he tells us. So the creation of wholes and totalities is of key importance. The Forms are wholes, and point to the totality at the root of reality.

 

It is possible to understand the consequence of the presence of this idea in Mesopotamia, because it is possible, with some teasing out, to understand the consequence of its presence in Greece. Which is that if the ground of Being is transcendent totality, then all other things which possess totality are in some way connected with it, and can be understood as representations of it. This is part of the logical basis for the creation of divine statues. But only part.

 

The transcendent totality is a plenitude, but it is ungenerated, and so it has no size. It contains the potential for all creation. It is one alone, because otherwise it would be other than it is. And yet Plato speaks of God creating a copy of Being, after which the world was patterned. If a copy of Being was created, then Being would be divided, not whole, and not what it is. It would have been subject to change, which is contrary to its very definition. And yet if it does not exist in the world of change, then it can do nothing. Movement and thought would be impossible. This would be a disastrous state of affairs for any cult which placed knowledge at the heart of Being.

 

Plato's argumentation about Being is unlike any other from Anselm onwards, in that he identifies Being with the nature of reality itself. Anselm and Descartes speak of supposed properties and attributes of the divine, but since they are attempting proof of the reality of God (framed in terms of some kind of existence), rather than proof of the realness of reality, there is an implicit and unaddressed separation between the divine and the nature of reality in their arguments. There is no sense in either case that the divine is absolutely dependent on the nature of reality.

 

Plato does not tell us how the paradox of one Being, who is necessarily two for the generation of the world of movement and change, and therefore not real Being, is resolved, nor does he resolve for us the question of whether or not the unchanging divine can be present in the world of movement and change.  What he does do is to proceed with the discussions, on the basis that it is necessarily so, that, somehow, the generated world is patterned after a representation of Being;  and that it must be the case that the divine can, somehow,  be present in the world of movement and change.

 

If we turn over in our minds the idea that the world is patterned after Being, without its essential nature being compromised, which is totality, alone and undivided, there is only one possible rational logical solution: that the world of movement and change is not, in reality, separated out from the nature of Being. What we see are aspects of Being itself, represented to us in various ways which are available to our capacity to perceive.

 

Language here does not serve us well, but essentially the logical conclusion must be that, if the essential nature of Being is to retain its integral nature, the world of movement and change is illusory. It has no reality apart from Being. Indeed, there is no such thing as existence 'apart' from Being: Being and the generated world are in some way coterminous.  The world we see and live in is a perception, no more and no less, contained entirely within unchanging Being, and composed of elements and aspects of Being. It is a perceptible world, rather than real; an immensely complex illusion generated within a reality which has no location, extension, shape or colour.

 

The consequence of this - and this is the crux of the matter - is that though Being retains its essential unchanging nature, and remains always purely is what it is, the generated world must have a double nature. That is, the world of generation is finite, and contains finite things, but it also necessarily contains things which are infinite. These things which relate to the infinite are the completions and perfections, and the other things which participate in the ground of Being. As finitudes, as conceivable images, they belong in the generated world as representations of the divine reality. Considered as representations of Being, they represent what we can understand of the divine, but they must necessarily also be Being itself, and be able to pass in and out of Being. Though we necessarily perceive these images as representations, unless we have profound knowledge of the divine.

 

It turns out therefore that we can add Plato to the list of ancient philosophers who were addressing reality as a paradoxical phenomenon. That changes quite a lot. It changes our understanding of the Greek mind, and our understanding of Greek culture.

 

We get some idea of the logic of divine images, their creation and worship, from this perception of reality and the world we live in as a paradoxical matrix. Things necessarily exist in two potential states. Secular and divine. Finite and infinite. Changeable and unchangeable. Sacred and Profane. These two states exist at the same time. It is a matter of understanding, which allows us to know that what belongs to the secular world, is also divine; that what is finite is also infinite; that what is subject to change also belongs to the realm of the unchanging; and that we can find elements of the sacred in the profane.  And, since the world of change is an illusion, so too are living and generated beings.

 

Formerly I've contrasted reality itself with the world of existence, which is, according to this way of looking at things, not truly real. It is sometimes useful to look at this contrast the other way round, in which the transcendently real world is the only one which has existence. This makes it possible to characterise the world of space and time as one which does not have real existence, only the appearance of existence to us.

 

However real things and people seem to be in space and time, they come to be and pass away, and do not abide. We think it has real existence of a kind because it has a form of consistency about it, in that physical laws exist and operate in it, and mathematics and geometry apply. This makes it possible to understand it in terms of a form of reality which has the property of objective existence, which is not dependent on mind. Mind observes it, according to this view, but it has existence apart from mind. 

 

Which brings us to the concept of necessity (anangke), which is key to the understanding of the natural world and the cosmos in Greece. Things which come to be and pass away are subject to it, as well as the objects which move in the heavens. Necessity refers to that force which determines how things behave which are not subject to the operation of the human will. So plants come forth and bloom, human beings are subject to divinely determined fate and destiny, and the planets move inexorably along their paths in the sky. The world of generation is characterised by the presence of necessity: it is a mark of things which have been generated, as opposed to those things which are aspects of unchanging Being.  But this property of the finite is necessarily determined by the infinite, as things which are finite are generated by it. They are patterned accordingly.

 

Greece shares with Assyria the notion that fate is all powerful, and that the destinies of both men and gods have been decreed at the moment of creation. In both cases however, fate and destiny are decreed by the transcendent divine, for the reason that the world of movement and change is an illusion contained within unchanging Being itself. It necessarily contains within it the start and the finish of all things which may take place in the secular view of reality. And Being determines these, according to its nature.

 

Maintaining this view of the division between the real world and the world of illusion, it follows that only the properties and attributes of Being are truly real, and truly existent. So, Being itself is real and existent, and its representations are also real, owing to their possession of its properties and attributes (by virtue of the fact that they are wholes and totalities, etc.). According to this view, things which have come to be are patterned after Being itself, and the Forms, which have divine properties as representations of Being.

 

In assembling lists of things which are pure, which have a common property, or a genealogy of the gods with their properties laid out as epithets which define their perfections and responsibilities, ancient scholars were also listing those things which they understood to be real and to have real existence, and which pointed to the place of creation. By contrast the world of generation contains mainly approximations to those truly real and existent things. These approximations to what is real however can be made perfect by skill, application and the pursuit of divine knowledge, and perfection and divine status can also be accorded to things and individuals by those who already have this knowledge.

 

By a short extension of this idea, it follows that those beings who live in the generated world have their true existence somewhere else. They are born into existence according to their destiny, and leave at the appointed time. Looked at from this point of view, death can be understood as a return to the place where the essential nature of the individual has its reality.

 

None of the foregoing implies belief of any sort. All of it may be teased out of the argument Plato makes about the nature of reality. This outlook is best understood as a doctrinal view, based on logical and philosophical discussion. It would have been subject to questions from neophytes and other philosophers; and would have been occasionally refreshed by interaction with those who understood similar doctrines from other places and cultures.

 

The normative view of religion in Israel for the first half of the first millennium B.C.E is that there was present a form of monotheistic culture and belief, engaged in a struggle with polytheistic ideas within Israel, and with Israel's neighbours in Mesopotamia. In fact we know almost nothing of religion in Israel in that period: there is no solid evidence which can give us reliable information. According to the redacted documents in the Old Testament, dating from the middle years of the  first millennium, it is clear that there was a protracted political struggle taking place for hearts and minds at some point, but most of the objections to polytheism seem to relate to experience of Mesopotamian religious cult and their ideas concerning the gods. Much of this valuable information may therefore post-date the Babylonian exile. 'Thou shall have no other gods before me' might be understood as a purely Israelite sentiment. However, 'I am that which is,' and 'I do not change,' (Malachi) references a philosophical notion of the divine, which we can now see is present in the context of the Mesopotamian divine pantheon.

 

Parts two and three of this work were written first, with the umbrella title of 'Being and Representation in Greece and Assyria.' I came to realise in the course of writing that it was important to make it clear that later discussions of the divine in the early modern period were addressing a much more limited set of concepts of the nature of the divine (which I have referred to as 'properties and attributes' more or less throughout), though these also represented an important subset of discussion of the nature of Being in Greece and Assyria. This early modern discussion represents a significant obstacle in understanding ideas about the divine in the ancient world.

 

Thomas Yaeger, 9^th April 2015.

 

At Reality's Edge

 

[Some notes I made while I was writing up The Mathematical Origins of the Megalithic Yard in early 2020. The notes conclude with some observations of the importance of the idea of limit in Mesopotamia, and its connection with the Assyrian Sacred Tree, and their notion of kingship.  I could have finished up with a short discussion of Egyptian interest in the idea of limit, particularly since we know (from the Rhind Papyrus) that they used the same method of calculation of Euler's number as in ancient Britain. That discussion with follow later.]

***

It has been twenty two days since I started to write up the article ‘The Mathematical Origins of the Megalithic Yard’ (mid February 2020). In this article, I suggested that those who designed the. circles came to the idea of the megalithic yard of 2.72 feet as the consequence of an interest in infinite series, and particularly those which approach a limit. The most important of these limits is the one which is known as Euler’s number, which, when rounded up from 2.7218… is 2.72.

This limit was first noticed in relatively modern times in the context of the calculation of compound interest, but the number, and the process by which it is arrived at, can be found in many other contexts.

Effectively, the number (when worked out to thousands of places), is the number as it would be found at infinity. So it can stand as an indicator of ultimate limit and of infinity. It is associated with the idea of ‘one’, as I’ve discussed in the article, and also as an irrational equivalent of one, which is a rational whole number.

An irrational counterpart to ‘one’, in a proto-pythagorean community, would have been easy to understand as belonging to a world beyond this one – i.e., a transcendent reality which is more perfect than this world, which is full of irrationality and measures which are incommensurable. The number may have been understood as being irrational to us because it is being represented in our finite world, and not irrational.

It also stood for the edge of our reality, and therefore would have signified the possibility of a joining between the transcendent reality, and our world of physical reality. Finding ways in which the worlds could be joined, and the incommensurate made commensurate, seems to have been a major preoccupation in the Neolithic, as it was also to philosophers and mathematicians in Greece during the second half of the first millennium BCE.

After I finished the article, I wondered how difficult it is to construct a series which will arrive at Euler’s number, how it might have been done, and how long it would take to come to the result.

A little research showed that there were many ways to construct suitable series of numbers, and a geometric calculation could produce a reasonable approximation reasonably quickly, without enormous calculations.  

 We don’t know for certain what base was used for calculations in the British Neolithic, but they were certainly aware of base 10, since they used powers of ten in their construction (ie, instead of a 3,4,5 triangle, they would sometimes use 30,40, 50 as their measures, knowing that the sides would be similarly commensurate after squaring). If they were using the English foot as their basic measure, it is likely they were counting to base 12 (ie, in duodecimal). But the construction of a series only requires whole numbers, arranged as fractions.

1 + 1/100000)^100000 = 2.7182682371923

100,000 is a lot of iterations, so it is unlikely that the determination was done in this way. The process will result in Euler’s number with any consistently generated series.

It can be done geometrically, which is much more practical, and is probably the technique which was used in the Neolithic. Using a sequence such as:

1/2  +  1/4  +  1//8  +  1/16  + ... = 1

Those who generated such a geometrical figure did so knowing that the series converged on a limit from observing the initial results. What they wanted was to find out a reasonably accurate value for the limit itself. The square could therefore be of any size (read as the value ‘one’), and might well have been created in a large field, with the fractions indicated by small stones.

I’ve written elsewhere about the importance given to limits and boundaries in ancient Assyria and Babylonia, particularly in connection with sites connected with the gods, and the rituals for the installation of the gods in Heaven. Sometimes aspects of the design of the Assyrian Sacred Tree were unwrapped, and represented on pavings as lotuses, alternately open and closed. Which is a way of indicating at these edge points that both possibilities are open, and even perhaps that opposing states are commensurate with each other in infinity.

It has already been identified that the Sacred Tree represents a form of limit, and consequently of the nature of divinity which has its true existence in a world beyond the constraints of finitude.  The design of the alternating lotuses also was used to separate the registers of images adjoining the collosal Lamassu statues which guarded the entrances of royal palaces. There was an image of the sacred tree, with two winged genies behind Assurbanipal’s throne, which seems to indicate that the king was understood to embody the transcendent reality which lies behind the world of the here and now.[the identification of the king with the divine reality appears in various royal letters] He is the perfect man, and the very image of God

[March 8, 2020]

 

[ Minor text corrections, Jan 1, 2021]