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.

 

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).

Pleroma, Cosmos, and Physical Existence

As suggested elsewhere, there isn’t much we can say about the initial state of physical reality at a notional time of its emergence, if the various parameters of what can be said of it don’t have any existence. That’s the problem physics has when it is looking for causes and mechanisms.

However it is possible to talk about the initial state of reality in terms of logical argument, which is how it was done in antiquity. They were familiar with talking about reality in terms of extreme states: does it exist? What is it? Is it one or two? If it is two is reality other than itself? Is reality complete in itself? Is physical existence a copy based on the pattern of reality itself? If it is a copy, has the nature of reality itself been compromised?

Whatever the initial conditions might have been, we can say that those conditions are at the edge of physical existence. Which is not to suggest that they actually occupy some kind of space at the edge of physical existence. Just that, since the initial conditions don’t participate in the conditions of our physical existence (extension, vectors, time, etc), then these conditions will, to us, appear to be something which we can find at the extremes of physical existence.

This is the root of the idea of the telos. It is about beginnings and endings – how things start and finish. When what is ultimately real is considered in this way, it is susceptible to logical analysis, and an idea of a prime mover beyond the properties of the telos itself is not required.

A discussion of this conception of the telos should be untarnished by the general deprecation of teleological argument in any kind of scientific analysis. We aren’t looking for purpose. But we are looking for the beginning, and how things might have unfolded from that beginning. The concept of the telos as a plenum, a pleroma undefined by the kind of parameters we find in our physical existence, as opposed to the idea that physical existence just appeared ex nihilo, can be discussed. Nothing as absence is very hard to discuss, except in the context of its opposite. In fact it cannot reasonably be conceived without that context.

The discussion of Aristotle (elsewhere) places his laws of thought into a wider context. The laws represent tools in the Greek dialectical armoury. So do the techniques used by Plato. But they are quite different and produce different kinds of argument and result in different conclusions. They belong to the same armoury (a discussion for another time). It is possible to understand some things with Plato’s approach which would not be possible with a rigorous application of Aristotle’s laws of thought. It isn’t the case that one logical approach is correct, and the other not. But they are appropriate to different contexts.

A plenum can be understood as identical with itself, and so, in that sense, can be thought of as consistent with the first of the laws of thought. But its properties, as understood from the point of view of physical reality, cannot be self-consistent, since it is beyond definition in physical terms. So the plenum must have a paradoxical aspect (literally meaning it is beyond human understanding), and therefore must breach the other two laws of thought (it can be one thing or its opposite; and it may also be neither one thing or its opposite).

There are several ways an argument about an initial and ever-present plenum can be taken forward. One is to take the view that physical reality represents a partial view of the plenum. Or can be understood as an assemblage of partial views of the plenum. It contains consistencies and regularities, but at a granular level (particularly), it behaves with apparent inconsistency, being best understood in terms of probability. That is how the plenum is, or at least the best way it can be understood by us. It isn’t one thing or the other. But occasionally its granularity looks like one thing or the other. And sometimes both at the same time. We can describe what is going on in terms of probabilities, which is how physics handles it, but it is not understood except in terms of mathematical description. The idea of the plenum, as established through a purely logical analysis, gives us insight into how the universe is actually operating.

It could be argued that physical reality behaves as it does at the quantum level because, for all practical purposes, the plenum has no size. So, at the quantum level, we are looking more closely at the nature of the plenum as it is, or rather as it must, on account of its nature, look to us.

Quantum entanglement might have a similar basis, on the ground that what is happening is actually happening in the plenum, rather than in physical space. Despite it having no size, it must necessarily be (in a sense) distributed throughout space and time.

In future I will be discussing Bell’s Theorem; Einstein, Podolsky, Rosen; and Klein-Kaluza. Also about the fact that Maxwell’s equations can be derived from Klein-Kaluza, and why the maths of Klein-Kaluza has two states.

Six Chapters from The Sacred History of Being

The first five chapters of my upcoming book are available to read by following the links below. A further chapter, which discusses George Berkeley's understanding of the Nature of Reality, is also available in the list.

Information about the publication status of SHB can be found among the Resource pages (right hand side of the blog pages).

[The Sacred History of Being was published by the Anshar Press on November 2, 2015]


Preface

Part One

A sense of the past
How old is Philosophy?
The Arrival of the idea of Being
The West and the Other
The Golem
Change and what is permanent
Recurring Questions
The Ontological Argument
The Ontological Argument in Anselm
The Ontological Argument in Descartes
The Nature of Reality in Berkeley
Hume and Kant on Reality

[Post updated on October 1, 2017]

The Arrival of the idea of Being

We can never recover the first instance when it occurred to someone that the character of living experience might be determined by the nature of reality itself, and that this exalted reality had a quite different nature to that manifested in the world of sense and experience. In other words, that the nature of reality was something quite other than that which could be determined by common sense. However it is possible to show that this kind of insight is not uncommon, and that it even can occur to a mind entirely unprepared for it by education, either in the present day, or in the past.

Many accounts of these odd insights into a different order of reality are presented in terms of mystical experiences, in which some aspect of the divine is encountered, or a strange exultation is experienced, or the entirety of the world is seen to be encompassed by a single grain of sand. But not all perceptions of the other nature of Reality present themselves in such poetic images.

This is a study of an idea: an idea which (I argue) underpins much of religious belief both in the present and in the ancient past, and the relation of religious ideas to the practice of philosophy, principally in the form of dialectical inquiry. This, expressed in one of a multitude of ways, is the idea of ‘the end’, the ‘telos’, of ‘completion’, and the nature of the ground of reality. The reality which simply is, and which does not move – and cannot move without breaching its integrity, is in essence the completion of completions, the end of ends, and was so understood. This is Being.

What is the essence of this theory of Being, from the point of view (first of all) of the historians of philosophy? Essentially they argue that it is a development of the concept of existence apart from its specific instances. That is, what it is to be, or to be real, in the abstract.[1] The appearance in text of the abstraction of the idea of Reality from specific, concrete instances, is read by modern scholarship as the first indication of the capacity for pure abstraction, so that for the first time it is possible to talk about things – concepts – which can have no specific instance, such as Being, Reality, Beauty, Justice, etc., but which, conceptually speaking, can be understood as lying behind specific instances. These questions are first clearly evidenced in the works of the Greeks: among the Presocratic philosophers, and most fully developed in the dialogues of Plato and the treatises of Aristotle.

This book does not deviate significantly from the contemporary scholarly idea that a theory of Being concerns existence apart from specific instances, and also key questions about the nature of reality in the abstract. It does however differ from the view of the historians of philosophy in that it does not read the first appearance of explicit discussion of the nature of Being and of Reality as the marker of its first discussion. Rather such evidence is just what it is: the first explicit discussion which we have available to us – created in a very specific set of circumstances. There is suggestive evidence that questions of Being were already current in the ancient world, and that the writings of the Greeks sometimes show their debt to these earlier sources.

Though framed as a historical study of its subject, this study is not intended to be definitive: we must walk before we can run, and such a history would be premature [2]. It is instead an exploratory survey of the cultural importance and pervasive scope of the idea of the end as a technical concept, a concept with a grammar and syntax of its own, which can be traced back (in iconographic, literary, and metrical terms) beyond the earliest direct documentary evidence. As a pioneering work, it is subject to errors of interpretation and detail, as all such works are. However, it would be disastrous for the progress of scholarship if the fear of making errors were to prevent the discussion of difficult subjects. I am sufficiently confident of the soundness of the general thesis to present it here, in a form refined by nearly thirty years of study.

By 1987 I understood that there was an important connection between the idea of kingship and completion in both Mesopotamia [3] and in Egypt [4], and was intrigued by the fact that completion was similarly an important idea in Greek philosophy, and, in the writings of both Plato and Aristotle, was also associated with kingship. Yet philosophy was supposed to be a Greek creation, without significant parallel conceptions in Mesopotamia and Egypt.

The importance of the concepts of completion and the limit in antiquity are inadequately explained by the scholarship of the past two centuries. The presence, the ubiquity of these concepts, reflected in some way at all levels of society, and across the whole range of ancient civilisations in the Near East and in Europe, implies the ubiquity of a teleological model of the world. In Greece itself, it is possible to trace the existence of this model at least as far back as Hesiod and Homer, which means the 8th and 9th centuries B.C.E., some four or five centuries before Plato and Aristotle, in whose works the concepts of completion and the telos form the backbone of many arguments.

For both Plato and Aristotle, the idea of completion is firmly associated with idea of the telos, and both ideas are involved in the definition of the Good, which is how these philosophers referred to the concept of absolute Being.[5] It is of course possible to accord importance to limits and ends without the presence of a teleological view of the world in support, and therefore the presence of a theory of Being. Even in such profoundly secular times we mark our passage through life with moments of explicit boundary crossing and transformation. However, where the terminology of limits and ends is used in a way which adapts the concepts to people, actions, objects, buildings, etc., then something more profound is involved.[6] Aristotle, for example, would speak of bricks existing for the final cause which was the house they complete; and Plato famously said that an individual existed for the sake of the end, and not the end for the sake of the individual. The context of this kind of usage, where these concepts form part of a coherent implex[7] of technical ideas, is what is of interest here.

A problem which has to be confronted however, is the axiomatic presumption of the scholarly, philosophical and theological communities that the concept of Being was not significantly understood before Socrates, and the assumption that it first receives forensic public discussion in the pages of Plato.[8] For instance, Henri Frankfort (et al.) in Before Philosophy, makes no mention at all of the concept of Being, or of a theory of Being, until the close of the book, where the legacy of the human intellectual adventure is being discussed. The index features three mentions of ‘Being,’ all referencing this final section. [9]

The reason why it is presumed that man thought about the world without a theory of being before the rise of classical philosophy is twofold – firstly, since the nineteenth century, there has been a general presumption that man has been refining his intellectual faculties, with the corollary being that the cognitive act of abstraction was less well developed in antiquity than it is now. Thus conceptions in antiquity were necessarily of a concrete nature.

Secondly, there is in existence apparently no theoretical discussion of theological matters (at least identifiable as such) of the kind which we find in the pages of Plato, or in the theologians of the Christian Church in the Middle Ages. This absence is taken, of itself, to indicate that in the early first millennium B.C.E. the philosophical discussion of theology, or matters related to theology, (such as the characteristics of ‘the One’, or ‘the Infinite’, etc.) was not practised at all, and that in all probability, was beyond the intellectual capacity of ancient man. [10] In short, the presumption of the presence of a theory of Being behind ancient discourse upon the world, in the form of myth ritual, symbol and icon, has no place in our approach to an understanding of antiquity. Not even as a hypothesis.

There is an important blind-spot in philosophical writings of the modern period however. This blind-spot can be characterised as an unwillingness or inability to address the possibility that reality might best be understood as a paradoxical matrix, in either contemporary discussion, or in earlier times. Writers in the 17th century say things like: ‘it is obvious that nothing can come from nothing’, and seem happy to rest on the assumptions of common sense. [11]

The essence of this common-sense approach is the assumption that the Aristotelian laws of logic – that a thing is itself; that a thing is either itself or something else; and that a thing is not something other than itself – are applicable in discussion or theorizing about the nature of the ur-reality underpinning the world of everyday experience, of description, including mathematical description, and that the world of everyday common sense experience and our understanding of it is in no way affected by the ur-reality and its properties. [12]

This is a strange assumption, and must seem so to anyone familiar with the religions and philosophies of the east which have their origins in antiquity. But the assumption is there. Plato's work does not enshrine paradox. We assume that we know this for sure because of Plato's demolition of the contradictions apparent in the argument of Parmenides about the One. If he demolished these arguments, he himself cannot have held such views about the nature of reality, in any shape or form. Yet he says very similar things in various parts of his extensive canon. He also explicitly says that certain characteristics of the forms allow them to 'pass into one another'. 

Pass into one another? It is obvious that this doctrine is not at all compatible with the Aristotelian laws of thought. Scholars have not been very alert to the fact that Plato is sometimes playful and mischievious by seeming to hold different views in different dialogues. It isn't simply,  as has been asserted, a matter of our lack of knowledge of the chronology and the development of his thought over time. He is trying to get us to think about what he is saying. What he appears to be saying is that, at one and the same time, ideas of the nature of reality are demolished by the use of logical argument, and yet these same ideas are nevertheless the case. 


***

[1] The study of Being is often referred to as ‘ontology’. It has two aspects – discussion of the nature of Being itself, and discussion of the ways in which we might understand the nature of Being. For those of an idealist tendency, it is often used to mean the study of the properties of Reality.  In modern times what we casually refer to as having existence (the raven, the cat, the table, etc.) have no real existence to the idealist, since these things do not (and cannot) have the properties and characteristics attributed to what might be referred to as the ground of true Being – the underlying Reality behind a world of appearances.
[2] Much of the evidence discussed in this book concerns the 1st millennium B.C.E. Professional historians and archaeologists are well aware that the history of the 1st millennium is a great deal less clear and certain than the lay public thinks, and therefore currently it would be extremely difficult to produce a satisfactory narrative account of the history of ideas in the 1st millennium B.C.E.
[3] The Akkadian term for king is ‘šar’, which also is a technical term indicating a completion. Herodotus was familiar with the Babylonian ‘Saros’ measure.
[4] Completion is a concept which appears in Akhenaten’s ‘Hymn to the Aten’, in connection wth both the birth of a chick from its shell, and the rising of the sun above the horizon on its daily voyage around the sky. Divinity is indicated in Egypt by the enclosure of the royal name in a cartouche, which is a representation of a length of rope, similar to that used to mark off an enclosure, a temenos, the foundations of a temple or city, or anything which has real being or existence.
[5] ‘Being’ was discussed as an abstract concept using the term ‘ontos’. All instances of Being were understood to be subsumed in the Good as the supreme end, for which all other ends exist. As such individual instances of Being were formal resemblances of the absolute Being upon which they depend for their reality.
[6] A modern illustration of the difference might be the way we refer to a head of state. We would have no difficulty referring to this person as the ‘head of his country’, or as ‘the political apex of his people’. We might also speak of him as ‘first among equals’, without quite understanding the origin of that phrase. But we would not speak of him as ‘perfected’, as ‘completed,’ or even 'the image of the divine.' Neither would we speak of him as the final cause of various actions and statuses in other individuals. Yet this is precisely how the ancients often speak.
[7] The use of the term ‘implex’ is borrowed from Giorgio de Santillana and Hertha von Dechend’s Hamlet’s Mill, Macmillan, 1969. The authors used this term to refer to a body of concepts related to each other, but difficult to separate in a sequential or narrative structure. Rather they related to each other in the way in which a series of variations on a theme might form part of a fugue.
[8] The same is held to be true of the idea of infinity. The reason we have any discussion of these ideas from Greece at all is the result of a rare separation of philosophy and religion in Attica.
[9] Frankfort, Henri, (Mr & Mrs); Jacobsen, Thorkhild; Wilson, John.A., Before Philosophy, Chicago, 1946 [original title: The Intellectual Adventure of Ancient Man]
[10] In the late nineteenth century and up until at least the last third of the twentieth century, this historic absence of the philosophic discussion of the divine was often seen within the frame of human evolution, so that the absence was explained in terms of our subsequent biological development toward a capacity to think in abstract terms. There never was any biological evidence for this, and no-one would mount such an argument today. Nevertheless, the idea of an intellectual development within historic time remains, at least as a cultural phenomenon. There was also a racist presumption that the Orient differed from the Occidental nations in their capacity to do philosophy at all. Within such a model of reality, there is both a disincentive to look for traces of philosophical thought in the ancient East, and an ever-present need to downgrade whatever traces are actually found. I recall a few years ago an Egyptologist stating that a symbol which had for some time been taken to indicate an unlimited number (i.e., the idea of infinity) meant no such thing, but rather a concrete conception of a limited (but very large) number. He was probably correct to say this. But it was clear that his purpose was to dismiss the idea that the Egyptians could have conceived the abstraction of infinity. The enlightenment notion of the primitive stupidity of ancient peoples took root before a single hieroglyphic text was deciphered, and it is at least a little odd that the judgements of scholars without access to a single word of ancient Egyptian are often borne out by subsequent research.
[11] John Locke is the case in point, who wrote this in his An Essay Concerning Human Understanding (1690); he regarded it as common sense that nothing could be generated from nothing. William Law expressed a similar sentiment. In both cases the assumption is that the Aristotelian laws of logic hold in a realm which is defined as transcendent of our own. This is a rather big and somewhat illogical assumption.
[12] Aristotle himself identified that insight involved the transcendence of a paradoxical situation, so that passing beyond an apparent contradiction resulted in understanding.