A Sense of Divinity - Descartes and Kant

The fourth programme of Adrian Moore's 'A History of the Infinite' (BBC R4, 2016) discusses the views of Rene Descartes in the sixteenth century, and also the views of philosophers from the eighteenth-century Enlightenment. I haven’t added up the number of centuries of thought which have not been discussed at all, but so far argument has been drawn from the sixth century B.C.E. (Pythagoras) fourth century B.C.E. (Aristotle, Zeno), the third century C.E. (Plotinus), the 13^th century C.E. (Aquinas), and the 16^th century C.E. (Bruno). Which is a journey of around twenty centuries. 

It isn’t that there is nothing to say about the idea of infinity during those long centuries, but that where Moore is going determined his selection of evidence and argument. He wants to talk mainly about the role and history of infinity in mathematics and in physics, and the fascinating paradoxes and problems which later investigation has thrown up. And a little about religious faith and the infinite. The first episodes are therefore a necessary introduction to set the scene.  

As he puts it in the text introduction to this episode, 'we have arrived at a time where people think about these things as we now do.' A telling statement, which hints at the richness and strangeness of the unexplored territory between the sixth century B.C.E. and the sixteenth century C.E., and that most of it is best skipped over as quickly as possible. It also lets us know that he has a normative view of human thought, and that what he thinks is rational and reasonable is mostly to be found in modern times. His is the Enlightenment agenda, which he mentions during this episode. 

Descartes famous ‘Cogito Ergo Sum’ (‘I think therefore I am’) is mentioned in the context of Descartes massive reduction of all the ideas and beliefs which he could accept unequivocally as true. He engaged in this reduction in order not to rely on tradition and authority, but on the intellectual resources available to the finite human mind. The question of whether the infinite can be grasped at all by the human mind is discussed, since we cannot see it or touch it. It is hard for us to know it, because it is the infinite. Descartes is quoted as saying that you cannot put your arms around a mountain as you can around a tree. So our knowledge of the infinite is necessarily less intimate than our knowledge of finite things.  

In the next part, the relationship between Descartes confidence in his own existence and capacity to think (expressed in the ‘cogito’) and his understanding of the infinite nature of God, is less than clear. It is true that Descartes suggested that he might have an idea of an infinitely perfect, infinitely powerful God because God put that idea into his mind. That might be the case. Alternatively, it may be that you as a finite being do not have to have an intimate acquaintance with the infinite in order to understand what you are talking about.  

Moore does not use the expression which Descartes employed to explain why it was not necessary to have intimate knowledge of something in order to have a useful and intelligible idea of what it is. He used ‘clear and distinct’ idea to indicate when he had such a useful and intelligible notion of what he was talking about. Later, Bertrand Russell would reformulate the distinction between knowledge by acquaintance and knowledge by description (in his Problems of Philosophy). So, by ‘clear and distinct ideas’ about God Descartes is relying on a description of what is, which means that he could be sure what he meant, and that his idea of God was a rational idea.  

In fact, Descartes idea of his own finite reality was dependent on his certainty of the reality of an infinite God. If he could conceive of such a God clearly and distinctly, then it was likely that such a God was real. 

Moore skips on to the second half of the eighteenth century, mentioning Berkeley (‘there is no such thing as the 10,000 part of an inch’ is all that is said), and Hume also, in connection with the indivisibility of reality (the disappearing inkspot when seen from sufficient distance, which is a matter of perception and experience rather than indivisibility per se). Berkeley was an idealist philosopher, who held that the only reason the world is perceptible is because it is held in the mind of God. He also denied materiality, at least as a metaphysical concept. 

Finally Moore discusses a narrow aspect of Kant’s understanding of the idea of infinity. This final part of the episode represents a highly misleading understanding of Kant. 

Moore argues that Kant agreed with Descartes that we have a clear idea of the infinite (the nearest he gets to the Cartesian formulation ‘things which are clear and distinct’). But that our idea is limited to what we can experience and perhaps what we can invest faith in. Really? I don’t think it is.  Did Kant say that knowledge is confined to the five senses? And if we don’t understand knowledge this way, we leave solid ground and end up in metaphysics? That is what seems to be suggested at this point in the series. 

One of Kant’s principal interests was metaphysics, and how we apprehend things and have knowledge of them. Hume’s empiricism was one of the things which impelled Kant to write some of his most important works (The Critique of Pure Reason, and The Prolegomena to any Future Metaphysics which may Present itself as a Science). It isn’t the case that Kant thought our ideas are limited to what we can experience in terms of the senses, but instead what is intelligible to us is interpreted through the categories of our understanding. He sought to understand shape and form without these things being associated with form possessing scalar values and spatial angles, which are matters of experience. In that he was very close indeed to Plato’s understanding of the Platonic forms. 

Kant, a figure so important to the concept of reason, is quoted as saying that ‘I go beyond knowledge to make room for faith’. It is true that Kant had the idea that rational thought and reason did not have to exclude a life of faith. It had space in which to exist. But it does not mean that Kant thought that faith was important to the life of reason. For Kant, like Pythagoras and Plato, knowledge is not gained through knowledge of sensible things, but is acquired by the contemplation of things which have a transcendent reality. This isn’t something which everyone can do, or will ever be able to do. Since there is an equation between the Divine and the Infinite, what Kant is doing is leaving space for some sort of understanding of the Divine for those who will never have a genuine understanding of transcendental reality and the Infinite. He is not arguing that faith creates a functional connection with the Infinite.

Karl Lōwith wrote that, in his book Religion within the Limits of Reason Alone, Kant had

interpreted the whole history of Christianity as a gradual advance from a religion of revelation to a religion of reason…. It is the most advanced expression of the Christian faith for the very reason that it eliminates the irrational presupposition of faith and grace.    

Moore then turns to Kant’s conception of the moral law. Aspects of the life of the mind which put us in contact with the infinite are about our reason, our rationality. Our reason enables us to grasp the moral law, which gives us infinite dignity (since we are rational beings). He says that “the moral law is what ought to direct us in all we do, with infinite respect granted to fellow rational beings”.

Which explains little. The origin of Kant’s moral law may be the idea that the life of reason, and rationality itself (as he defined it) is about connecting with the infinite. If man is truly rational, then he is connected with the Infinite (the ancient concept of the soul, as discussed by Plato, is related to this idea). But we need to accept Kant’s understanding of what reason is, and not distort it by saying knowledge is obtained through the five senses. Through this distortion, what Moore is left with is the Calvinist notion of a ‘sensus divinitatis’ (sense of divinity).  Which is a poor substitute for the kind of engagement with divinity which was understood to be possible in the ancient world. Such engagement was not achieved through knowledge of the world of the five senses or space and time.

Plato and the Transcendental Infinite

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

That is exactly the doctrine of the ascent and descent via the Forms which is described by Plato. The definition of transcendent reality in Plato (articulated by Socrates) is that it is a place beyond shape, form, size, etc., and occupies no place on earth. It is however the place where knowledge has its reality (the ‘eternal and incorporeal things’ mentioned by Pythagoras). Connection with transcendent reality is possible by the likenesses to the transcendent which have existence on earth, such as things which are complete and whole, which therefore participate in the completeness and wholeness of the transcendent reality. Completeness and wholeness require (in the world of the mundane) delineation and limits, and so the limits and the extremes of things are also things which participate in transcendent reality.

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

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

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

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

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

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

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

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

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

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

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

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

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

Adrian Moore on Georg Cantor and the Size of Infinity

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

The '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 Wider Scope of Ancient Mathematics (letter to an American Scholar)

 

Avebury Circle, photographed in 2001

Dear....., 

Hi. I became aware of your short book [.......................]  relatively recently. I wish I’d known it earlier.

I have a strong interest in the idea and function of the concept of limit in antiquity. My main object of study at UCL was ancient  Assyria (mostly the text corpus). Like the Greeks, they had a strong interest in the idea of limit, which is illustrated on the walls of their buildings, and is also represented in their images of the sacred tree. Limit also serves an important function in setting up their gods in heaven (I’ve written about both Assyrian and Babylonian rituals for this).

This tells us something of the actual basis of Mesopotamian religion, which has an origin which is quite different from what we imagine. 

Essentially ancient religions are transcendentalist in nature. In other words, they have their origins in a focus on abstract conceptions (limit, infinity, infinite series,completion, totality, etc). Which makes a nonsense of the idea that the Greeks were the first to grapple with sophisticated abstract thought. Clement of Alexandria created a list of civilizations which practised philosophy, and added the Greeks as the* last* to adopt the practice of philosophy.

Since you might be interested in the wider scope of ancient mathematics, I am writing to you to point you at a couple of articles which illustrate that these concerns were a feature of building projects in Neolithic Britain also. The Horus numbers are there, as the basis of establishing Euler’s number via a geometric construction. Euler’s number being the final result of a convergent infinite series.

Did they get their mathematics from Egypt, or did they develop them themselves? I have no idea. Why Euler’s number? It’s a mathematical stand-in for the extreme limit, which is infinity.


‘At Reality’s Edge’

https://shrineinthesea.blogspot.com/2020/12/at-realitys-edge.html?spref=tw%20%20# (Short article)

‘The Mathematical Origins of the Megalithic Yard’

http://shrineinthesea.blogspot.com/2020/02/the-mathematical-origins-of-megalithic.html (Long  article)

Best regards,

 

Thomas Yaeger

At Reality's Edge

 

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

***

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

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

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

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

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

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

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

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

1 + 1/100000)^100000 = 2.7182682371923

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

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

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

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

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

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

[March 8, 2020]

 

[ Minor text corrections, Jan 1, 2021]

Mathematics and Calculation in Antiquity (letter to a Cambridge Scholar)

Date: Sun, 06 Dec 2020 12:56:04 +0000
To: .............cam.ac.uk
From: thomas yaeger 
Subject: Mathematics and Calculation in Antiquity

Dear........,
 

I’m supplying here the address of an article which may be of interest to you, since a) you are interested in early examples of sophisticated human cognition, and also b) in examples of ideas, languages and cultures being transmitted west to east in deep antiquity. This article addresses both of these areas.

The article took seven years before it assumed its current form. It started off as a relatively minor component in a project on the presence of abstract ideas in the ANE and the Levant before the Greeks, which resulted in my book, ‘The Sacred History of Being’ (2015).

What the article argues is that the mathematics which can be found in the vast majority of megalithic rings in Britain, France and elsewhere, show that builders had a grasp of infinite series and Euler’s number from very early on (late 4^th mill. BCE onwards, up until around 1400 BCE, which is when they seem to have stopped constructing them).

The pattern of their distribution around Europe and the Mediterranean suggests the original builders travelled westwards, and then north to Britain.

One of the reasons why no-one has considered the presence of Euler’s number in these structures (2.72, supposedly first discovered by Bernoulli), is of course, why would they know this number? It is also assumed that the number would have been too hard to calculate in such ancient times, even if they did have a loose grasp of infinite series.

This is not actually the case – it can be established geometrically with a relatively small number of iterations (less than a dozen). Interestingly the procedure for doing this can be found in the Rhind Papyrus, which dates from around the 17^th century BCE, but was originally compiled earlier. In a publication issued by the British Museum in the late eighties, Gay Robins and her husband identified that the Egyptians were working with an understanding of infinite series. And showed the Egyptian diagram, illustrating how it was done.

The geometric process for establishing Euler’s number can be done on the ground, using small stones. I explored the Avebury complex pretty thoroughly in 2001 and 2002, and noticed  brickish sized stones collected together, on the edge of one of the circles, almost lost in the grass. I had no idea why they might be there at the time, but they may have been what they used in the geometric construction . Effectively, the small stones are telling us what the whole structure is for.

 

The article is at:

http://shrineinthesea.blogspot.com/2020/02/the-mathematical-origins-of-megalithic.html

The short book on the Rhind Papyrus is at:

https://www.amazon.co.uk/Rhind-Mathematical-Papyrus-Ancient-Egyptian/dp/0714109444

My book is available from CUL (and elsewhere) in eBook format. 

[text correction, December23, 2020]  

 

Best regards, Thomas Yaeger.

December 6, 2020.

On Infinity and Creation (Seven Discussions)

 

Featuring Richard Dawkins, Lawrence Krauss, Arthur C. Clarke, Carl Sagan, Stephen Hawking, Roger Penrose, Carlo Rovelli, Helen Czerski, Adrian W. Moore, John D. Barrow, and H. Peter Aleff.

 

'Something from Nothing' A narrow understanding of reality plus a limited knowledge of the history of ideas can make idiots out of otherwise really bright people. A conversation with Richard Dawkins & Lawrence Krauss. youtu.be/q0mljE9K-gY via @YouTube #RichardDawkins #LawrenceKrauss #SomethingFromNothing #Atheism #Cosmology #Physics

 

‘God, the Universe, and Everything Else’ (1988). A discussion (52 minutes) with Arthur C. Clarke, Carl Sagan, and Stephen Hawking. Moderated by Magnus Magnusson. #God #Religion #Spirituality #Science #Physics #Mathematics #Infinity #Mandelbrotset #Time https://www.youtube.com/watch?v=HKQQAv5svkk

 

‘Why Did Our Universe Begin?’ (Roger Penrose) A short discussion which shows the limitations of trying to understand creation purely on the basis of mathematics and physics, and a great deal of inference. #Infinity #BigBang #Creation #Entropy #RogerPenrose #Universe https://www.youtube.com/watch?v=ypjZF6Pdrws

 

Physics in all its glory (BBC ‘Start the Week’, first broadcast November 9, 2020) Nobel Laureate Sir Roger Penrose joins Carlo Rovelli and Helen Czerski to discuss black holes and ocean currents with Andrew Marr. https://www.bbc.co.uk/programmes/m000p6dl #Physics #Cosmology #Creation #BlackHoles #Size #CarloRovelli #RogerPenrose #HelenCzerski

 

Evading the Infinite: a Review of Adrian Moore’s radio series, 'A History of the Infinite', broadcast by the BBC in 2016. #Infinite #Infinity #philosophy #Reality #Mathematics #Being #SetTheory #Plato #Aristotle #Plotinus #Neoplatonism #Hilbert https://shrineinthesea.blogspot.com/2017/10/obscured-by-clouds-critical-review-of.html

 

Continued Fractions Professor John Barrow (Gresham College lecture) https://www.youtube.com/watch?v=zCFF1l7NzVQ transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/

 

The Egyptian Shen Ring. Discussed at Peter Aleff's 'Recovered Science' site. An Egyptian exercise in abstract and philosophical thought. #Shen #Egypt #Abstraction #Philosophy #Transcendentalism #Infinity #Nun #Atum #Creation #Numbers #HPeterAleff http://www.recoveredscience.com/const104shenring1.htm

Jump Cut: The Pursuit of Knowledge, God and Reality in ‘2001: A Space Odyssey’

Clarke and Atheism

The more times I see 2001: A Space Odyssey, the stranger the movie gets. Arthur C Clarke said a couple of things about the film which illuminate what it is about, and a number of details in the film offer further clues. Other work he became interested in much later, also casts light on the real subject of the film.

Firstly, there is Clarke’s famous statement that Kubrick and Clarke had persuaded MGM to fund an enormously expensive religious movie (he actually meant a theological movie, but such a distinction might have been lost on the moguls of the time). His books are in fact often peppered with ideas which approach theological questions, yet he described himself as an atheist many times during his life. This apparent contradiction needs to be explored.

A good overview of the complexity of his views on both religion and theology can be found on the Wikipedia page on Clarke’s life:

Themes of religion and spirituality appear in much of Clarke's writing. He said: "Any path to knowledge is a path to God—or Reality, whichever word one prefers to use."^[105] [Mintowt-Czyz, Lech (19 March 2008). "Sir Arthur C. Clarke: The Times obituary". The Times. London. Retrieved 6 August 2008.]

And:

He described himself as "fascinated by the concept of God". J. B. S. Haldane, near the end of his life, suggested in a personal letter to Clarke that Clarke should receive a prize in theology for being one of the few people to write anything new on the subject, and went on to say that if Clarke's writings did not contain multiple contradictory theological views, he might have been a menace.^[106] [Clarke, Arthur C. (1999) [1991]. "Credo". Greetings, Carbon-Based Bipeds!. First appearing in Living Philosophies, Clifton Fadiman, ed. (Doubleday). New York: St. Martin's Griffin. pp. 358–363. ISBN 978-0-312-26745-2. Retrieved 8 January 2010.]

I think Haldane was right about how dangerous some of his ideas were, and that he often contradicted himself on matters of theology outside the scope of science. The following illustrates how he occupied different intellectual spaces from early on in his life, and right up to the end:

 When he entered the Royal Air Force, Clarke insisted that his dog tags be marked "pantheist" rather than the default, Church of England,^[43] and in a 1991 essay entitled "Credo", described himself as a logical positivist from the age of ten.^[106] In 2000, Clarke told the Sri Lankan newspaper, The Island, "I don't believe in God or an afterlife, ^[107] and he identified himself as an atheist.^[108] He was honoured as a Humanist Laureate in the International Academy of Humanism.^[109] He has also described himself as a "crypto-Buddhist", insisting that Buddhism is not a religion.^[110]

So he was characterising himself as a pantheist at the time he joined the Royal Air Force in WW2, but in 1991 he says in ‘Credo’ that he was a ‘logical positivist’ from the age of ten. Is this possible? It’s a contradiction, but I think it is possible that both characterisations are true. He was a mathematician and a scientist, which doesn’t preclude an interest in profound questions about the nature of the universe and reality, which are less amenable to purely rational answers. We will come back to the ‘crypto-Buddhism' later.

In saying that he was a pantheist in his early twenties, I think he was indicating that he was already making an equation between theology, the divine, and the nature of reality itself. Those of a mathematical bent sometimes do, since the mathematics of the physical world reveal something of how reality works behind the physical representation of it. But if you are going to investigate reality itself through mathematics, you need to stick close to the evidence. To that extent he was a logical positivist for the whole of his life.

So Clarke was interesting in theology and theological questions. But he clearly distinguished between those questions, and the principal territories occupied by modern religions:

A famous quotation of Clarke's is often cited: "One of the great tragedies of mankind is that morality has been hijacked by religion. ^[110] He was quoted in Popular Science in 2004 as saying of religion: "Most malevolent and persistent of all mind viruses. We should get rid of it as quick as we can.” [Cherry, Matt (1999). "God, Science, and Delusion: A Chat With Arthur C. Clarke". Free Inquiry. 19 (2). Amherst, New York: Council for Secular Humanism. ISSN 0272-0701. Archived from the original on 3 April 2008. Retrieved 16 April2008.]

Yet Clarke was happy to engage in dialogue with those who were not locked into a view of religion which saw faith as its core. Alan Watts was one of those:

In a three-day "dialogue on man and his world" with Alan Watts, Clarke stated that he was biased against religion and said that he could not forgive religions for what he perceived as their inability to prevent atrocities and wars over time.^[112] Clarke, Arthur C.; Watts, Alan (January 1972). "At the Interface: Technology and Mysticism". Playboy. Vol. 19 no. 1. Chicago, Ill.: HMH Publishing. p. 94. ISSN 0032-1478. OCLC 3534353. ]

Alan Watts of course, was heavily influenced by Buddhism, which Clarke said was not a religion. That distinction is an important one. Buddhism is a way of approaching reality, which assumes that everything is (in some way) related to everything else, both in terms of representation, and In terms of causality.

Despite his atheism, themes of deism is a common feature within Clarke's work.^[[115] (20 March 2008). "For Clarke, Issues of Faith, but Tackled Scientifically". The New York Times. ISSN 0362-4331. Retrieved 21 January 2020.]

Edward Rothstein understood the deeply rooted dichotomy in Clarke’s approach to understanding the nature of the universe. Buddhism of course famously managed to construct a theological understanding of reality which did not much require discussion of gods, which is one of its most attractive features. And in case anyone was in doubt about Clarke’s seriousness about that kind of atheism:

Clarke left written instructions for a funeral that stated: "Absolutely no religious rites of any kind, relating to any religious faith, should be associated with my funeral."^[116] "[Quotes of the Day". Time. 19 March 2008. Archived from the original on 24 March 2008. Retrieved 20 March 2008.] 

Crypto-Buddhism

Clarke spent more than half his life living in what is now Sri Lanka (he moved there in 1956). He appreciated the good diving opportunities available in the warm seas around the island, which represented the nearest experience to the weightlessness of space he was likely to experience in his lifetime. Sri Lanka  was also a relatively cheap place to live. Writers then as now found it difficult to make a decent living out of their writing, so moving there made practical sense.

Clarke’s closest friend was a Sri Lankan, who he met while he was studying in London in 1947.(Leslie Ekanayake).Their association lasted for the next thirty years, until the premature death of Ekanayake. So Clarke is likely to have had discussions about Buddhist ideas on the nature of reality long before he made the decision to move to Sri Lanka. His understanding that Buddhism was a way of engaging with the nature of reality which was, despite appearances, not a religion, may have been acquired from discussion with Ekanayake.

Clarke refer to his engagement with Buddhist thought as crypto-Buddhism because he read the body of ideas contained in Buddhist thought differently from others. He saw Buddhism as a way of attempting to understand reality in philosophical terms, which also allowed the possibility of exploring reality with mathematics and geometry.

Buddhism is a body of ideas which, like many religions in the east, embraces paradox, and the importance of what cannot be seen. What is on the surface, is not all that there is. Investigation of what is puzzling about reality is required in order to gain understanding, and ultimately, enlightenment. I was given a small statue of the Chinese goddess Mu when I was in my twenties, made from peachwood, which represented her as holding a lotus above her head. Mu represents the all, from which everything is made, and what is made is what floats on top of the waters. But though the lotus emerges into visibility, it is not itself the All. It is connected with it (the statue hold the lotus flower by the stem), but is just a representation of what lies unseen in the waters.

I’ve described some aspects of the Buddhist approach to what is hidden, and the Buddhist understanding of causal processes, elsewhere (‘The Enlightenment of David Hume’). For the early Buddhists (I’ve written about the scholarly issues around the antiquity of Buddhism in ‘The Age of the Buddha), the ideas that reality itself is hidden from us, and that how things are represented to us depends on causal relationships which are not necessarily obvious, clearly depended on a sophisticated philosophical model of the world. One of the reasons for the importance of scholars and priests in Buddhism is that thought and actions are required in order restore balance where balance has been disrupted. Everything is understood to be connected to everything else, and is understood to be a cause of something. Since we do not have direct and unmediated access to the invisible all, careful investigation of these issues by those who have a profound understanding of them is required.

So what were Clarke’s actual views of God and the nature of Reality itself? Clarke kept a journal during the writing and production of 2001, which gives us some clues [Clarke included some of this journal in his book ‘The Lost World’s of 2001’. The journal has been quoted elsewhere also]. At one point he records a discussion of Cantor’s theory of transfinite groups with Kubrick, without going into any detail, or giving a context for such a discussion. Transfinite groups gave Cantor a great deal of intellectual and psychological difficulty, because of the implications (that you can have infinities which are different sizes, but they are all infinite, for example. Which again implies that all things are connected with each other, and each thing shares the same identity).

It is likely that Clarke was expounding something of his mathematical view of the nature of God and of Reality to Kubrick. This was the way he understood that theology had to work, and both faith and belief had nothing to do with theology.

It is clear that he understood God and Reality to have some profound relationship to actual infinity (as opposed to an Aristotelian ‘potential’ infinity). Modern scholars (both mathematicians and theologians) ignore actual infinity on the grounds that (they think) it is impossible to work with the concept. This doesn’t mean it makes no sense to address the question of the actual infinite as Cantor did. Clarke had the actual infinite in his mind, since he referenced Cantor directly in his conversation with Kubrick, and didn’t just confine himself to the mathematics involved in the theory of transfinite groups. The nature of the world in which we have our existence bears some relationship to the actual infinite, rather than the hollowed out version of the infinite which is subject to mathematics and geometry in space and time.

The teaching machine which appears to the man-apes close to the beginning of the film was not the first choice of object to serve that purpose. But it is the object that Clarke and Kubrick settled on. The reasons for this choice are interesting on account of its dimensions, which are precisely outlined in the novel associated with the film (and elsewhere). It is a black oblong block, whose dimensions are one, four, and nine units. That is, one squared, two squared, and three squared. That is the beginning of an infinite series, which, if extended, would eventually reach infinity itself. There were discussions about what images would be displayed on the monolith to the man-apes, but Clarke and Kubrick decided not to show any of these, or even explicitly suggest (in the film) that the monolith was communicating with the man-apes. However the dimensions of the monolith, embodying the beginning of an infinite series suggest that the communication was emanating from the infinite itself.

Jupiter and Beyond the Infinite

The Stargate sequence in the film begins after David Bowman’s struggle with HAL (and his purely logical and algorithmic artificial intelligence, which results in HAL’s  murder of the crew who were still in hibernation), and once they are in Jupiter space. Jupiter is of course the king of the Gods (Clarke’s  book locates the Stargate near Saturn). During that sequence David Bowman’s space pod travels over an abstracted landscape: he is travelling somewhere, but it clearly isn’t in real space. At one point, seven double tetrahedrons appear, hanging above the landscape. Each of the tetrahedrons is filled with geometric lines which are in motion. Each of the tetrahedrons contains the same geometric patterns, which change in perfect synchrony. This image is very reminiscent of Leibniz’s description of the monads which he posited were the foundation of reality. All of the Leibnizian monads reflect each other, in both nature and in processes. All of them are derived from the principle monad, which is the foundation of Reality itself [Leibniz was a student of Chinese philosophy and oriental patterns of thought, as well as a polymath and logician].

Why are these images there in this part of the film? Douglas Trumbull, who was responsible for many of the special effects in the film, has said that the images in the double tetrahedra were built from reprojections of the moving slit-screen generated landscape below the tetrahedra. Which by itself doesn’t tell us very much, except perhaps that both the landscape and the monads were meant to be different representations of the same thing. One shows an abstracted representation of travel through space; the other shows mathematical and geometrical change which might not exist in space at all. This is likely to have emerged from suggestions from Clarke, but I am not aware that such conversation is recorded. But it can be understood as a product of Clarke’s self-declared Crypto-Buddhism.

These ideas may have their origin not via Leibniz, but directly through Hindu and Buddhist texts. One of the most relevant ideas is that of Indra’s Net.

Indra’s Net

Quoting Wikipedia once again, at: https://en.wikipedia.org/wiki/Indra%27s_net

"Indra's net" is an infinitely large net of cords owned by the Vedicdeva Indra, which hangs over his palace on Mount Meru, the axis mundi of Buddhist and Hindu cosmology. In this metaphor, Indra's net has a multifaceted jewel at each vertex, and each jewel is reflected in all of the other jewels.^[5]

In the Huayan school of Chinese Buddhism, which follows the Avatamsaka Sutra, the image of "Indra's net" is used to describe the interconnectedness of the universe.^[5]  Francis H Cook describes Indra's net thus:

Far away in the heavenly abode of the great god Indra, there is a wonderful net which has been hung by some cunning artificer in such a manner that it stretches out infinitely in all directions. In accordance with the extravagant tastes of deities, the artificer has hung a single glittering jewel in each "eye" of the net, and since the net itself is infinite in dimension, the jewels are infinite in number. There hang the jewels, glittering "like" stars in the first magnitude, a wonderful sight to behold. If we now arbitrarily select one of these jewels for inspection and look closely at it, we will discover that in its polished surface there are reflected all the other jewels in the net, infinite in number. Not only that, but each of the jewels reflected in this one jewel is also reflecting all the other jewels, so that there is an infinite reflecting process occurring.^[6]

The Buddha in the Avatamsaka Sutra's 30th book states a similar idea:

If untold buddha-lands are reduced to atoms,

In one atom are untold lands,

And as in one,

So in each.

The atoms to which these buddha-lands are reduced in an instant are unspeakable,

And so are the atoms of continuous reduction moment to moment

Going on for untold eons;

These atoms contain lands unspeakably many,

And the atoms in these lands are even harder to tell of.^[7]

Book 30 of the sutra is named "The Incalculable" because it focuses on the idea of the infinitude of the universe and as Cleary notes, concludes that "the cosmos is unutterably infinite, and hence so is the total scope and detail of knowledge and activity of enlightenment."^[8] In another part of the sutra, the Buddhas' knowledge of all phenomena is referred to by this metaphor:

They [Buddhas] know all phenomena come from interdependent origination.

They know all world systems exhaustively. They know all the

different phenomena in all worlds, interrelated in Indra's net.^[9]

How old are these ideas? They are a lot older than Greek ideas about the infinite, and the idea reflected to us from the 1^st millennium BCE in Greece that Reality itself is necessarily One, which was a question which Plato mentioned as of key significance to our understanding of Reality.

It is worth noting that the section caption film refers to ‘Beyond the Infinite’, rather than just 'The Infinite'. This I think would have been a formulation by Clarke, given his understanding of Buddhist ideas. The infinite is incalculable and ineffable. We can say it is unbounded and without limit, and so on. But describing what it actually is, is another matter. The first translator of the works of Plato, Aristotle and  the Neoplatonists into English, Thomas Taylor, wrote about this question, and the Greek interest in it, at the cusp of the eighteenth and nineteenth centuries. Though it remains a question which is not often (if ever) discussed in classes devoted to philosophy or classics.

The Three Million Year Jump Cut

It is possible to understand the film version of 2001 as a film with a broken back. Kubrick was in charge of the script for the film, and Clarke was writing the novelisation. They talked together and shared ideas of course, but Kubrick had a different idea of how the film should be. I’ve quoted evidence of Clarke’s philosophical interests. Kubrick did not share most of these, hence the fact that, in the course of production, Clarke talked with him about Cantor’s ideas, which he knew nothing about.

In the end, Clarke’s understanding of how the film should be was very different from Kubrick’s, so what we have as the final product is actually a collision between two quite different perspectives. Kubrick’s general view was that nation states (i.e., organised societies) had always behaved like gangsters. There was very little good to say about them. His earlier films bear this out: ‘Paths of Glory’, ‘Spartacus’, and ‘Dr Strangelove’. None of which paint a picture of a species which is keen to avoid war, destruction and casual killing. He still felt that while he was making 2001, and his later films (‘Clockwork Orange’, ‘Barry Lyndon’ ‘’Full Metal Jacket’) suggest he retained something of that world-view for most of the rest of his life. So the beginning of 2001 features the struggle for survival of a group of man-apes, eking out a precarious living in the dry African savannah millions of years ago, in the vicinity of a contested water hole. They eat vegetables and are prey to carnivores. Their prospects are not good.

Then one of the man-apes has the idea to use an animal bone as a tool, and by extension, a weapon. Everything changes. They have access to better nutrition, and gain hegemony over a competing group of man-apes by beating their leader to death, and as far as Kubrick is concerned, the future is set. The implication is that the idea was first suggested to the man-apes by the monolith.

Cue the jump cut to orbiting space weapons. The implication is that nothing of significance has changed over the intervening millions of years.

As if that is the human story. What does this do to the movie? It means there is no space available for anything which happened in between - culturally and intellectually. None of that is of any importance to this story. Clarke could not have included much about early human intellectual development in the west, but he could have included material from the east.

The consequence of inserting this jump cut is that, though the development of the human race is perhaps to be conceived as being  towards a grasp of infinite knowledge, and an engagement with Reality itself (Clarke’s understanding), there is no space in this film for reflection that this is an old idea, and that human beings were aspiring to this over many thousands of years, east and west. As I’ve indicated, there are residual clues in the film that a more sophisticated view was discussed in the early days of the production.

Instead, Kubrick peddles the rather lame idea that human evolution will take us to infinity, with the help of those unseen beings who first installed the monoliths in various parts of the solar system. Despite the fact that it seems in Kubrick’s view, the evolution of the human species just intensifies a meaningless struggle for survival. Bigger weapons, and ever more violence.The unseen beings were the ones who encouraged the use of tools and weapons, and now, at the end of the line, David Bowman has mysteriously reached infinity in any case, and is reborn as a divine being.

2001 is a deeply unsatisfactory film, when it is examined in detail. It makes it much harder to explain human cultural history, mainly because that cultural history is just swept away by Kubrick as of no importance, in one 25^th of a second. 

[Retitled September 21st, 2020]