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 'Hill of Many Stanes'

[An extract from a conversation with a correspondent in the US, from May and June, 2020, shortly after 'The Mathematics of the Megalithic Yard' was completed.] 

On Monday, June 1, 2020, 09:31:47 AM PDT, Thomas Yaeger [....] wrote:

[....], hi. Thanks for your mail. I'm going to respond to it in separate mails, since there is a lot to say. Interleaved, as usual (bad academic habit!)

At 06:03 29/05/2020, [....]  wrote:

Hi Thomas,
Sorry I haven't responded sooner. I've been working on a response to your article (& other emails) about the Megalithic Yard and didn' t want to write again until I had made some progress. I'm probably making it into too much of a project lol. [....} So, I'll send what I have for now (including other stuff I've been putting in a draft) and get it off to you. Sorry if it doesn't do justice to your arguments. 

[....] 

Your argument is very compelling and interesting. It seems like a real breakthrough although, naturally, I'm not enough of an expert to judge!

I think it is a real breakthrough, but it took a while to make it (as I said, the article was written in about a day and a half, after thinking it through for around two years). Developments are happening very fast now, which is interfering with my writing programme.

. I understand that math as such isn't the point of your argument, it's more about what Euler's number signified, right?

Yes. It's Euler's number, what it represents, and how they calculated it in the 2nd and 1st millennia BCE. I think I've changed my mind about how much Alexander Thom actually knew. I think he knew that it was a pointer towards the idea of the infinite. But he did not know that in those ancient days the ideas of the divine, the infinite, and reality itself were regarded as coterminous, and were just different ways of speaking about the same thing (which is an understanding which still survives in Hindu thought and religion). So for Thom, he could see the mathematics, but didn't understand the idea of reality itself as a primal fulness, or a plenum, and why that would engage ancient interest.

There is in Scotland a site near John O'Groats which is known as 'the hill of many stanes', which has remained uncleared since the neolithic. In the documentary he says he is impressed by what the builders of the circles were able to do without pen and paper, and logarithms. But that without such constructions (as the 'hill of many stanes') 'you can't really do it'.

What was he talking about in this short insert into the documentary? He doesn't explain what the small stones were for, or how they were used. I think I understand now that the field of stones was used to calculate Euler's number, in the context of an engineering construction. That site needs extensive re-evaluation.

Thom's book publications are very plain and not dogged with interpretation. I think he realised that what he could do, and get away with, was to draw attention to the fact that something very interesting and mathematically disciplined was happening in the Neolithic and Early Bronze Age, but the whole thing was just too big a pill to swallow for the academic community. He held back.

One thing that interests me is people's motivations, in particular, which of their psychological needs are being served by engaging in different courses of thought and action. I assume that people have always been curious about life and the world (some more than others, of course!), but what struck me about what you wrote is people's need for or a sense of order and structure in order to feel a degree of safety in a world that is challenging to fathom.

 It depends on where you are in society. Sometimes, as now people are told convenient lies (there is no money!), or circumspect evasions. Ancient priesthoods, because of their picture of the world, understood themselves to be dealing with the nature of reality itself. Neophytes would be chosen from all levels of society, since it was necessary to put a premium on intelligence, in order to join the worlds and make the incommensurate commensurate. Reality itself was the home of all knowledge, and all possibility. You can't deal with that without intelligence. The rest of society would have to make do with what Plato described as likelihoods, because they were too far from an understanding of reality.

Thom was not a classicist or a historian, so he did not know (as most modern scholars still don't) that ancient religion was about *knowledge* (scientia). The ancient priesthoods understood themselves to be dealing with knowledge, and that their activity was a science. That's all changed, but we continue to project modern religious intellectual weaknesses into the ancient past.

Thanks for the photographs.

More later,

Best,Thomas

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.

 

Reality and Perception in Plato's Academy

[a letter from June 3, 2019, written to a scholar interested in how reality was understood in the ancient world before the Greeks. There is quite a lot of evidence for that understanding in existence, in philosophical texts from the classical world, and also in literature and art from Greece and elsewhere. The clear commonalities present in ancient iconographic evidence have scarcely been addressed so far - present in Egypt, in Mesopotamia, in ancient Anatolia, in Europe, and beyond. But they cannot be interpreted on a purely common sense basis. So they are, for the most part, not interpreted at all.

In The Sophist, Plato presents us with a fundamental conundrum concerning the nature of reality, and how that reality is to be understood. The argument is framed as a logical one. But the key to it is altogether left out. Instead we are told that we must accept that it must be true that both movement and change, and the unchanging nature of the One, coexist in what reality is, otherwise we would be faced with choosing one or the other (essentially the argument of Heraclitus or the argument made by Parmenides). That is an impossible choice.

The logical basis of this argument can however be figured out, provided we jettison some assumptions along the way. Discussions by Plato across his dialogues offer some clues...]

***

I think you will be interested in this paper (link below). I remember analysing the structure of Plato’s The Sophist in 1994, but over time, I forgot about the argument it contains, or even that I’d made one. I batch-scanned a lot of paper documents in 2003, and the analysis of The Sophist was one of those. But I didn’t read it again until recently.

The original document is squibbish, was written quickly, and was never properly completed or edited. But, knowing what I now know, I’d found the essential arguments for the ancient priestly understanding of reality, all collected together in one literary work, without being entirely aware of the implications of that. It is a little eerie to read this document now, since it looks far beyond what I was sure of at the time.

What Plato is doing in The Sophist is what he did in many other dialogues (not all), which was to include reminders to those who had been trained in theological doctrine what was important, and to wrap this information up with more or less irrelevant speculation for the merely curious and uninitiated.

The discussion of the four outlooks on the nature of reality which feature in The Sophist represent discussions which took place in the ancient equivalent of the seminary (it is odd that we don’t have information about the existence of these institutions in ancient Greece, unless the Academy was exactly that). The importance of the discussion is that it establishes that the Real is essentially and necessarily paradoxical.

There is the idea of the One, and there is the experience of the many. If there is only the One, there is no life, movement or thought. If the many are real, then it is difficult to understand how there can be something like the One, which retains its nature, and abides.

Not everyone who participated in these discussions would have become a priest, because not everyone would have settled for position b), which, to some casts of mind, would have seemed to be deeply unsatisfactory. But acceptance of position b) is the one the priestly establishments were looking for in their candidates.

Why position b)? It is suggested in the course of the dialogue that it has to be accepted, in order to account for both our intellectual understanding of the nature of reality, and our experience of the world of movement and change.

That however is not a philosophical argument. Something is being glossed over at this point, and we have to look outside The Sophist to understand that. The answer to this problem is Plato’s concept of The Good, articulated by Socrates in The Timaeus.

Socrates said that the ultimate reality, whether it be termed the form of the Good, or given another necessarily inadequate name, does not reside in space - in fact the notion that things [which are truly real] have a place is described as

"a kind of bastard reasoning": we dimly dream and affirm that it is somehow necessary that all that exists should exist in some spot and occupying some place, and that which is neither on earth nor anywhere in the Heaven is nothing.

In The Phaedrus Socrates speaks of the region above heaven,

“never worthily sung by any earthly poet". It is, however, as I shall tell; for I must dare to speak the truth... the colourless, formless, and intangible truly existing essence, with which all true knowledge is concerned, holds this region and is visible only to the mind...

Which means that Socrates is referring to the concept of the plenum: the reality we experience a partial representation, a slice, of what is contained in the totality of what is possible. If the plenum itself is possible, then the experience of change and motion is also possible. But as a perception.

The point of position b) is that it recognises the paradoxical nature of reality, and that what is represented to us is a subjective representation of Being itself. There is only Being, and the experience of physical and secular existence is a partial view of what is contained in the plenum. We see what we see, but it is not reality itself. It is what we can see and understand.

Is this a purely Greek understanding? I think it isn’t. Pythagoras (according to the Neoplatonists) spent around twenty years in Egypt imbibing their doctrines, as well as having discussions with priesthoods in the Levant and Mesopotamia, while in the service of Cyrus. Some of that went into Plato’s work, according to the Neoplatonists, though there is also strong evidence (which I’ve discussed) that ideas familiar to Plato were already present in archaic Greece.

The blog page which points to the paper (‘Magic or Magia? Plato’s Sophist’) is at:
 http://shrineinthesea.blogspot.com/2019/06/magic-or-magia-platos-sophist.html The link to the file is at the foot of the page. The article has its own DOI, and resides at the Zenodo archive (CERN).

An Appetite for Knowledge

I have argued elsewhere that ideas of Being, of the nature of reality, and the divine, were once approached  in terms of conjectures about the reality (or otherwise) of the one and the many. These conjectures follow on from the initial question, which is: why is there something rather than nothing? Plato’s argument, following on from propositions made by Parmenides, who declared that we should look only to the one, and that only the one truly is real, is the most sophisticated of all discussions in antiquity concerning why there should be something rather than nothing.

Plato argues that we should always look to the ‘one true thing’. This is different from saying only the one exists, or only the one is truly real.

The anthropologist and classicist J.G. Frazer was very dismissive of Greek questions concerning the one and the many, saying that they constituted ‘popular questions of the day’. The argument of Parmenides about the nature of Being  remained entirely undiscussed by Frazer.  But then he argued that questions concerning Being were entirely barren, since nothing could be predicated of Being.*1

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. We like to see Plato’s articulate discussion of Being as the surfacing of a human capacity to grapple with abstract ideas, and the marker of our emancipation from irrational ideas about the world and the gods. For Frazer, Plato was as guilty of intellectual error as any of his contemporaries, as well as his predecessors.

In late Hellenistic times, there seems to have been a very poor grasp of the context of the development of philosophy around the Mediterranean. 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 this was so in the Protagoras, saying that the practice of philosophy was of a great age – perhaps contemporary with the arrival of peoples from Egypt, who settled in the Peloponnese, and also in Crete. He also presented Solon in discussion with Egyptian priests in the pages of the Timaeus, who found the Greeks to be very young, and not conversant with knowledge ‘hoary with age’.  

Aristotle presented the common sense view that philosophy was first developed in a place where there was a leisured class, with the time and resources to think about philosophical questions. He may have had Egypt in mind, since Egypt had professionalised priesthoods. Later philosophers such as Porphyry suggested that key parts of Pythagorean doctrine came west to Greece from Babylon, in the late sixth century B.C.E.*2  Plato references details of this doctrine, without connecting it explicitly to Pythagoras.*3 Aspects of that doctrine can be found elsewhere in the pages of Herodotus (concerning Solon),*4  and also in Homer’s Iliad (Book 18), where a number of key details associated with the doctrine are run together in close order, without being explained. *5

The former Priest of Bel at Babylon, Berossus, moved to Athens, and wrote about Babylonian history and philosophy, describing their system of knowledge as based on the idea of an initial plenum (which is a philosophical concept) using the image of a sage emerging daily from beneath the sea (symbolising primal fulness and abundance), granting knowledge to man about the sciences, agriculture, and the practical arts. *6 He also gave information about the Babylonian New Year festival (Enuma Elish), the liturgy for which contained accounts of two creations: the first irrational, and the second one, rational, and which was given its rational character by the chief of the Babylonian gods.*7  Berossus' book was the Babyloniaka, unfortunately now lost. But the christian writer Eusebius had access to it as late as the early fourth century C.E, and quoted a number of passages from it which survive in his writings. Recovered Mesopotamian texts, including tablets containing the Enuma Elish liturgy, generally confirm the accuracy of Berossus. 

Not much of this was of use to the Enlightenment agenda, which preferred to look at the development of philosophy in Greece as the first beginnings of a rational understanding of the world. And so the information was deprecated and ignored. The phrase ‘I doubt that’ is a dangerous one in the classics community. It is a way of saying ‘this is not the consensus view of scholars and the profession’. Usually no discussion follows, since the opinion is usually an opinion of the worth (or otherwise) of evidence. Scholars weigh evidence, and they do so (they are convinced) with better tools than were available to ancient scholars. The judgement is fitted to modern requirements. So, as a result, it is clear that it is unlikely that Solon visited Egypt, and that Pythagoras visited Babylon. Tread carefully, or your credibility as a scholar may be in doubt.

Thus, the scholarly consensus is that philosophy is an autochthonous development. The culture of the modern west rests on this idea.Why in Greece? The idea of the ‘Greek genius’ won’t cut the mustard any more, at least by itself, but I have heard the phrase uttered by people who should know better. But during the high days of the enlightenment, and the beginnings of what became the fully-fledged discipline of Classics, that is what the scholars wanted. Something pure and out of the orbit of other cultures, which, by definition, had no philosophy or anything which would measure up to something like rational thought.

Sometimes history is built backwards. It isn’t just a matter of looking to the historical record and starting from that. History always has been in part about critical scrutiny of sources and judgements, even among the Greeks *8 But as Bernal pointed out in the first volume of his Black Athena, the critical revision begun in the enlightenment was wholesale, and created an alternative representation of the origins of European civilisation.

That was the agenda. To reinterpret the past, and in terms of a rational and enlightened understanding of the world. Hence, the philosophe Denis Diderot wrote not just about ideas and philosophy in his Encyclopédie, but also about the arts and crafts. The latter may have been wrapped up with myth, folklore, and superstition, but they were still essential to the rational life of man, so these were also added to the Encyclopédie. Everything from the past which served some purpose, or which could be made to serve some purpose within the rational enlightenment model of reality, was critically examined, and reworked to fit what was intended to become a new understanding of man and his place in a new world of reason. A new understanding of how man might live.

I wrote in the article ‘Logical Modality in Classical Athens’ that there are in fact two logical modalities present in the writings of Plato and Aristotle. Aristotle is mostly (though not always) concerned with the modality which has come down to us as the basis of formal logic. Plato is clearly aware of this modality, but, though no modern academic has dared to identify the other modality as logical, it is. It is simply that to us, it does not describe relationships which should be described as logical. Plato  talks about this other logical modality several times, in the Republic and in the Timaeus. It is connected with the doctrine of wholes and totalities, and is the basis of explaining how things may participate in other things, which is not a pattern of ideas which fits with Aristotle’s general understanding of logic.*9

Why is this important? Simply put, it matters because it is the basis of the Greek understanding of how transcendent reality relates to secular and physical existence, which was a matter of great significance up to the middle of the first millennium B.C.E, and beyond.  The doctrine underpins the understanding of the divine as something which can be both transcendent, and immanent, and thus be present in existence. *10 *11 It also points to a rather strange conclusion about the nature of the reality in which we live and think.*12

It might be imagined therefore that this doctrine would be the subject of a great deal of scholarship. In fact there is very little on the subject. The dialogues in which the doctrine appears have been written about endlessly over the last two centuries, but, though Plato’s discussion is noted, the fact that his discussion is based on an alternate logical modality is not acknowledged, and the conclusions which might follow from treating it as such, do not follow, and are therefore not discussed. It is treated as it is often presented by Plato – as mathematical and geometrical metaphors for how things might possess some form of congruence with each other.

I've argued elsewhere that there is, in general, very little appetite for attempting to understand Plato in his own terms.*13  When he talks about transcendent reality, this is treated as some sort of literary fiction, which has no necessary properties of its own. When Plato talks about the Forms, this also is treated as a species of literary fiction, which Plato records as being demolished in his Parmenides and in the Sophist. When Plato discusses the soul (in the Timaeus), it turns out that it is something which has the property of being connected with the Form of the Good, and so he argues that knowledge is acquired by the activation of that connection. We have forgotten what we knew (apparently) through the shock of physical birth, but it is possible for us to reacquire this knowledge by looking to the one true thing.*14 

Deriving all knowledge from the Form of the Good is also seen as a bafflingly impenetrable notion to the modern mind, since Plato talks about ascending purely in the mind from Form to Form, and then descending back to the world of physical reality, Form by Form, and says this is the only way to acquire genuine knowledge. How can real knowledge be acquired in this way? Why should Plato argue like this?  How can any of this make sense to us?

It makes very little sense to us, because we have lost the original doctrinal context of his discussions, and we are not that interested in attempting to recover what we can of that context, even for the purposes of a better scholarly understanding of what he is talking about. So the study of Plato languishes lifeless in the seminar room, taught and discussed from generation to generation by people who have no clear idea of what Plato meant. Books and papers are published, which clear up a minor detail here, discuss another one there, but do not leave us much the wiser.

We can understand the range of existing discussion about Plato in terms of what scholars do not or cannot understand about Plato, and their attempts to fit what they think they understand about his work into some kind of modern frame. It is their minds, and the categories of their own understanding which create the problem, not the obscurity of Plato’s ideas.

One of the principal reasons we cannot easily understand Plato is down to the loss of an understanding of that alternative logical modality. That understanding needs to be restored, along with a basic comprehension of why it is important. Not just for our understanding of Plato himself, but also for our understanding of his cultural context; the context in which philosophy was understood to be of inestimable value; and also for an understanding of some very strange things about the ancient world, which are all the stranger because (it seems) they made sense in antiquity (sacrifice, divination, idolatry, prophecy, omens, oracles, etc.).

One of the most valued books in my library is Religion and Magic: Approaches and Theories, which is a survey of modern anthropological thought. *15 The author is Graham Cunningham, a specialist in the ancient Near East and the relevant languages. Anthropology is a relatively young discipline, though crowded with many points of view. *16 Cunningham’s book covers the whole range of these, at least in terms of summarising the views of those who first suggested those theoretical approaches. He divides the approaches into several sections, which are:

1 German pioneers, 2 Early Intellectualist approaches, 3 Emotionalist approaches, 4 Phenomenological Approaches, 5 Structural Functional Approaches, 6 Symbolic Approaches, 7 Recent Intellectualist Approaches, 8 Structural approaches, 9 Cognitive Approaches, 10  Feminist Approaches.

All of these approaches were developed and used without any meaningful distinction being made between ancient cultural phenomena and cultural phenomena of modern times. I write carefully here, since there are unpleasant presumptions in the discipline of anthropology, which have not yet been rooted out. Anthropology as a formal subject was founded in the early nineteenth century, and the presumptions of the time are necessarily locked into the work of the pioneers. Many of these can be called into the light of day if you scratch the modern anthropologist in the course of discussion. The fact is that a dangerous equation was early made between the cultures of antiquity and the world of  'the primitive and the savage' in the modern world (which the Classicist D’Arcy Wentworth Thompson referred to as ‘running folklore to the death’).*17

So Cunningham’s book covers two centuries of thought about culture, civilization, religion, magic and ritual. All premised on the assumptions, understanding and categories of knowledge of those living and working in those two centuries. Nothing about those matters is covered from earlier centuries. It is as if the study of human culture, human thought, and the nature of man himself, began only in Hegel’s study, and nothing of worth came before Hegel. *18 

I could digress here, and lay out what came before in detail, but that is for another time. I will allow myself to say that Plato had something else to bring to the party, which is not covered in Cunningham’s survey; the Neoplatonists (who thought of themselves as Platonists, but we will not let them be what they are) would have brought the same thing to the party, as would some of the early Gnostic writers. The Platonists of the Italian and English Renaissances also understood what Plato was writing about, at least for the most part, and would be shocked that, not only do we not understand Plato, but that we have chosen to explain much of human culture in terms of a fundamental conceptual error about the way reality works, and with a complete disregard for the way the human mind was once understood to engage with that reality.

It is assumed by the moderns (in the west, at least) that philosophy is a cultural phenomenon which, in historical terms, follows on from religious thought, and does not precede it. We are sure of this because we are sure we know what religion is, and what philosophy is. We think that religion enshrines a form of knowledge about the world and the universe which is less than rational, and that the devotees of religion are often credulous. Religious thought is often transfixed by the numinous and the invisible, and its responses to these things are unfathomable to the rational mind. We also think religion, to a great extent, developed in response to the ideological needs of social groups, sometimes in conflict with other social groups, which can give very distinctive structures to the myths and rituals which become part of the bedrock of the public show of these religions.We believe this (admittedly oversimplified) picture to be a correct interpretation of what religion is. Modern religion differs from ancient religion only in details.

Whereas philosophy is about a critical and rational understanding of the world and how it presents itself to us. Religion is cast as an un-philosophical and irrational understanding of that reality. The development of philosophy represents progress.

These notions are subject to nuance of course, but this is the core of the enlightenment idea of what these things are.

We assume that religion, as it presents itself to us in the form of the relatively young Abrahamic religions, is not fundamentally different in nature to other religions, either elsewhere now, or in the distant past. We make such an assumption, because there is nothing in our understanding of other religions which suggests to us, at least not with any clarity, that any other process is active in the creation and operation of religion. The dynamics are the same, and the functions are the same.

Graham Cunningham’s postscript makes it clear that religion is generally understood by anthropologists to be a relic of an earlier time where rational thought was absent. Certainly in terms of what we would recognise. Somehow religion is still with us, because large parts of the human race believe in the importance and efficacy of belief in the divine: that belief in the reality of the divine is something which is essential to the religious life, and the religious understanding of reality. *19

Yet this is not what ancient writers tell us. They tell a different story, as I’ve suggested, and one which is not represented by any current anthropological approach. That different story is that religion in the ancient world was not about belief, but concerned both knowledge and conjecture about the nature of the divine; and about the nature of reality itself.*20

I prefer to write about ancient religion in terms of divine cult. I distinguish the two for clarity - religion is often understood in the modern world in terms of sociological and ideological functions. And indeed religion has these functions: it often provides social cohesion, ideology, and useful rules of social behaviour. But to attempt to explain the core of divine cult in such terms is presumptive. If the core of a divine cult involves rational conjecture about the nature of reality itself, we will not easily find this out, even if the evidence is screaming this at us. In antiquity the ritual practice of divine cult was often a private affair, rather than a public one, which implies an important division between the performative function of ritual, and the outward show. Ritual and observance were the important things. The public show might echo the thought behind the cult in some respects, and offer clues to its significance and nature, but it is not the cult itself. *21

The association of knowledge with the core divine cult (and, consequently, a confirmation of the  connection of the cult with the divine itself), gave the rituals of ancient religion (both private and public) their efficacy, and patterned their meaning. But this did not involve belief.  Belief is something which is not supported by credible argument, or is a step beyond the limits of an rational argument.  That is not what was involved in ancient discussion of questions concerning the divine. Can you have knowledge without rational modes of thought? And what did the ancient writers mean by knowledge of the divine, and by equating knowledge with the divine? Clearly, it is not possible to have actual knowledge of something without rational modes of thought. But what rational thought is, is another one of those things which we are quite clear about in modern times, even if the evidence points to a different conclusion, and exists in teeming quantities.

___

 1. Frazer wrote this in his prize essay of 1879 on the 'Growth and Development of Plato's Ideal Theory'. Published finally in 1930. I'd found this essay after noticing that there were odd features in the structure of The Golden Bough. I wrote about this in J. G. Frazer and the Platonic Theory of Being (2016), and also in the more compact essay 'Frazer and the Association of Ideas', published in Understanding Ancient Thought (2017).
2. In his 'Life of Pythagoras' Porphyry tells us that Pythagoras travelled extensively around the Near East and Egypt in the service of Cyrus. Discussed in 'Pythagoreanism, the Divine, and the Nature of Eternity'.
 3. In the Timaeus. Discussed in 'The Platonic Theory of Being' (full text with notes). Originally published in The Sacred History of Being (2015).
4. Discussed in 'Solon in the Court of Croesus' (extract). The full text is a chapter in The Sacred History of Being (2015).
5. Discussed in 'Working Wonders: Hephaestus and the Armour of Achilles'. A retitled chapter from The Sacred History of Being (2015), originally titled  'Being in Homer'.
 6. Discussed in 'Oannes and the Instruction of Mankind', which is one of the appendices to The Sacred History of Being (2015).
7 Discussed in 'The Babylonian Creation': an extract from the chapter 'Creation' in The Sacred History of Being (2015).
 8. Modern scholars are well aware of this. Ancient historians arranged information, but also questioned it. The Assyrians seemed to have a similar attitude to their materials - the chronicles associated with Ashurbanipal's reign exist in more than one version, with events recorded in a different order.
 9. Discussed in 'Logical Modality in Classical Athens' which is a chapter in Understanding Ancient Thought (2017).
10. Consequently this ought to be a matter of interest to Christian theologians, owing to the importance of the idea of the incarnation of the divine in the Gospels. For centuries the incarnation has been presented as a mystery (to both believers and theologians), without any kind of theoretical basis. But the theoretical basis is present in ancient literature, in various places. It even surfaces in a text in the Nag Hammadi documents.
 11. The logical basis of the incarnation is discussed in 'The Keys of the Kingdom: Binding and Loosing in Heaven and Earth', a chapter from Echoes of Eternity (2020).
12. Discussed in 'The Transcendental understanding of Reality', and in 'The Greek Ontological Model in the 1st Millennium B.C.E.'  from Echoes of Eternity (2020).
13.  'The Sweet Song of Swans'. The full text of this chapter, which concerns (among other things) the modern creation of the Greeks as the originators of philosophy, was published in the The Sacred History of Being (2015).
 14. In the chapter 'Plato's Theory of Vision', from The Sacred History of Being (2015). This extract concerns the discussion of the cosmos as “a perceptible God made in the image of the Intelligible.”
 15. Cunningham, Graham, Religion & Magic: Approaches & Theories, Edinburgh University Press, 1999.
 16. That there are so many conflicting views might suggest that perhaps few, or even none of the approaches are correct, but are simply the product of the failure to grasp the sophistication of ancient thought.

17. Discussed in 'Running Folklore to the Death', an extract from section 13 of J.G. Frazer and the Platonic Theory of Being (2016).
18. I've discussed Eriksen and Nielsens' A History of Anthropology (2001) in the chapter 'Before Anthropology', which book illustrates that modern anthropology is sometimes about how things ought to be, rather than a profound engagement with the evidence. They do however give a useful account of what came before Hegel. Published in Man and the Divine (2018).
 19  We tend to inject our presumptions about the past onto the evidence, as if common sense is a servicable tool in understanding it. It takes more than that. Discussed in 'Beyond the Religious Impulse'. Published as a chapter in Understanding Ancient Thought (2017). See also 'Distinguishing Belief and Faith', in Man and the Divine (2018).
20 A close look at Assyrian divination reports makes it clear that there was nothing mystical or vague about Assyrian queries about the future. The queries to the Sungod (Shamash) were expected to provide accurate information in response. This is because the diviners and scholars formulating the questions understood the universe to be rooted in a plenum. In which case,  all things which might be, already exist in the plenum. The diviners were therefore asking the Sungod for information which was available to the god. Twenty divinatory queries are available on the page 'Who will Appear before the City?' The concept of the plenum and its connection with the Mesopotamian story of the creation is discussed in the chapter: 'The Idea of the Plenum in Babylon'; further discussion of  the concept can be found in  'Pleroma, Cosmos, and Physical Existence', both from Understanding Ancient Thought (2017).

21. Discussed in 'Shar Kishati and the Cult of Eternity', published in Understanding Ancient Thought (2017).

Draft version, October 29, 2019. Final version, All Hallows Eve, 2019. Updated 11 January 2020. TY.