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.

Heidegger and Being: (Writing to @SemprePhi)

Hi. I read some of Heidegger’s Being and Time when I was around twenty, not realising that it was only the first third of a much larger projected volume. His essentially existential approach to gaining a theoretical grasp of ‘Being’ left me cold. This is because even then I didn’t think there was an existential approach possible, since, by definition ‘Being’ by itself is beyond actual existence.

Or at least by my definition, which was fuelled (at the time) by the sort of thing Plato said about it, and some of the statements by the presocratics. We might imagine that it exists, as all things must do, somewhere in space and in time. This Plato said was an illegitimate way of thinking about ‘the one true thing’. By its nature it cannot have its reality and nature in space and time.

Heidegger is aware of this of course, but nevertheless deflects from the consideration of Being as Being (the presocratics Aristotle described as the first philosophers to consider the question of Being qua Being). He looks leftwards again and again at existential and phenomenological approaches to questions of Being.

In the end, he argues, in a manner similar to Sartre later on when talking about meaning (in Existentialism & Humanism, which is the slightly inaccurate title his book is given in Britain), that existential experience, however imprecise and however poor it is as a process,  is the way in which to gain an understanding of what is difficult to apprehend.

In short, for me, Heidegger is not actually talking about Being at all, but the Being of living beings, and occasionally looking upwards at more abstracted and transcendent conceptions of what Being itself might be, though it is an unattainable concept.

Heidegger was a pupil of Edmund Husserl, who has been a major influence on me, also, though it a different way. Many of my essays on subjects start off from one place, and end up in another. Occasionally I follow the line of an established author or authors, and as the essay goes on, I start to deconstruct the elements which hold their argument together. After that, I may entirely reverse the premises of their argument, and go off in another direction. I was doing this before I had my close encounter with Husserl, but learning to see all information as elements which might be capable of being read and constructed in entirely different ways, and breaking down conventional narratives into their components, was a key part of my intellectual development.

I also had the advantage of having read widely in philosophy and ancient literature before I began formal study of the ancient world. So there was much more going on in my head than a direct interaction with texts as themselves. There were always other possible contexts, and other ways of understanding what their actual concerns might have been (this widened to an interest in iconography and ritual texts – no one treats the latter seriously (in their own terms), outside of a purely anthropological interest. Wittgenstein thought this was a foolish approach by the anthropologists).

One of the first things which set me on my current path was noticing (as I’ve mentioned elsewhere), that some equivalent technical terms in ancient Egypt were used in the same way as Aristotle did. But the Egyptian evidence was nearly a thousand years older (the concept of ‘completion’ for example). So it occurred to me that both the established history of philosophy (i.e., invented by the Greeks in the 5^th century BCE), and that the idea that philosophy was the successor to religious thought (considered scarcely rational), might be components in a false narrative. So I started to move the building blocks of the conventional narratives around, to see if they made more sense when looked at from a different point of view.

I’ve mentioned Aristotle referring to the presocratics as the first philosophers to consider Being as Being, meaning in all the abstracted glory of the concept (Aristotle made an important distinction between the actual and the potential infinite, and the latter is still the basis of most mathematical work on the infinite. Why? It is hard to do mathematics and geometry with the actual infinite. The modern assumption is that nobody before the Greeks ever conceived of a direct engagement with the infinite (or with Being).

Yet the prosocratic concern with Being qua Being is in the wrong place. If it was in the right place, it would follow on from the kind of discussion we get from Plato and Aristotle, when contemplating various intellectual puzzles (the aporia). Yet there it is at the beginning of the Greek philosophical tradition. It is in fragments only, many of which were used by Aristotle in his Metaphysics. Could it be that these fragments were the product of a sophisticated discussion of philosophical questions in the Eleatic school and elsewhere, much earlier, and that such discussion has just dropped out of the record? Or perhaps was never committed to writing at all?

I know the answer to that question now, which is that the Greeks did indeed get their philosophical ideas from elsewhere, which is not to say that they did not do philosophy themselves before this (I’m in the process of writing up the evidence and case for the Greek close encounter with Ancient Assyria in the seventh century BCE.)

I’ve been reading an extensive paper on Heidegger, and his uncompleted project of Being and Time. The latter half of the proposed book was actually looking to a deconstruction (or even destruction) of the history (and modern understanding) of philosophy. That was a big surprise, since it is essentially what my overarching project is about. For me, it is largely a necessary task because the Enlightenment placing of the first sophisticated discussion of philosophical questions in Greece between the seventh to the fourth centuries BCE entirely obscures the relevance of philosophy to the history and development of religion, both in the ancient Near East, and in ancient Greece. That false narrative needs to be fixed.

I’ve beaten over a thousand words in about an hour, so I will break off here. If you are interested the essay (actually seminar notes) on Heidegger’s Being and Time is at: https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/8/1250/files/2016/01/Heidegger-Being-and-Time-1qtvjcq.pdf . By Karsten Harries, from 2014.

I’ll write about Parmenides and Heidegger’s discussion of Plato’s Sophist later.

Hope you have had a great start to the year!



Best, Thomas

What We Have Lost, and How the Ancient Concept of the Cosmos Died

[Extract from correspondence with a specialist in ancient astronomy, on the question of why it is we are so disengaged with the concerns of the ancient world.] 

[....]

Thanks for your substantial mail, which I will work through in order, for clarity.

The day of the equinox was quite spectacular, with the full moon rising over the Pentland Hills to the south of Edinburgh. The weather has been very unsettled in the past couple of weeks, so I was not expecting to see anything.

[...]

...binding and loosing was important in Egypt also. In fact it seems to have been an idea universal in ancient civilisations. Though the basis of the idea is likely to have been the preserve of the educated, and the personnel of the divine cults (mostly, but not always exactly the same people). The core idea underlying binding and loosing is that our physical existence is, however it might look to us, a subjective perception of reality. And a subjective perception of reality is a subjective perception of Being itself.

That is why it is so difficult for scholars to understand what is going on. They see the practices and ‘beliefs’ which survive in the literary and archaeological record, but they have no frame in which to understand these things. They do not expect to find anything subtle and sophisticated, because they (they think) are dealing with forms of primitive stupidity, no matter how refined the cultural remains might appear to be.

The figure I gave for the number of offering tables recently turns out to be well out of date. The archaeological exploration of ancient Akhetaten (Tell El Amarna) continues, and the current figure is more than 790 offering tables. That is a staggering number.

You can term the perception of the cosmos which largely eludes modern scholars ‘a shamanic world view’. It is a world view which is extremely old. There were always people around who could see and think beyond the here and now. But the term ‘shaman’ is relatively modern, and fills a gap in the arsenal of modern anthropology. But supplying a name for a phenomenon does not explain what is behind the shamanic practices, and the associated patterns of understanding.

Once the inextricable link between the physical world and the reality beyond became part of the ancient mindset (probably from an immemorial time), it becomes obvious that what happens in Heaven has an effect on Earth, and vice versa. This explains a lot about Buddhist thought, which I’ve spent some time studying over the years. Recently I wrote an article on the possible influence of Buddhism on the philosophy of David Hume. The article explores the logic involved in Theravara Buddhism in some detail.  Theravara Buddhism isn’t a close analogue with with Mesopotamian and Egyptian thought, but it arises from a similar perception of what Reality itself is. The article also discusses ideas prevalent in the European Enlightenment (mainly in the context of Diderot’s Encylopedia, which explain much about negative attitudes to ancient thought in the eighteenth century). The Enlightenment of David Hume.

How was the ancient perception of the nature of the cosmos lost, and why? That’s two very large questions! Asking about the consequences of that loss is another big question. You know lots of stuff about that already, I know. But a discursive triangulation on this question between us, is likely to be useful for us both. This may take some time.

I’ve been writing about that loss for a long time, since it became obvious early on that it was necessary to understand the trajectory, the context, and the mechanics of the loss (in some detail), if we were to be able to (properly) understand what was actually lost.

As far as the savants of the European Enlightenment were concerned, the entire history of the human race before the eighteenth century was a morass of irrational thought – mythology, folklore, magic, legions of superstitious beliefs, a general credulity in the human race, and in earlier times, murderous ritual. They thought they could get clear of all of that by the rigorous application of common sense. That was the basis of Diderot’s Encyclopedia. So, the enoblement of common sense is one of the factors in the loss of the ancient world view.

It is hard to entirely blame the savants, given the limited information that they had. To them the past seemed to be a grim and dark place. They understood that even the crafts were wrapped around in superstition. But the crafts remained important for the wellbeing of the human race. So the crafts were described in the Encyclopedia, but shorn of anything which wasn’t entirely practical.

The savants often did not read and know ancient literature. So they did not know much that was worth knowing. That’s one of the problems with common sense: if the solution is common sense, that is already in your head. Reading is irrelevant. 

Common sense of course, when you are dealing with something which is transcendent of human experience and understanding, is entirely inappropriate. Common sense dictated that the transcendent did not exist, and served no purpose in understanding our world.

The loss of understanding has been going on for a very long time, and was already under way in the 2^nd Millennium BCE. This island is dotted with stone circles as you know, but the ancient Britons stopped building them around the 14^th century BCE. So something, some conception was lost, which removed the motive for building them.

I read a chapter of your writing about the esoteric recently, which was essentially sound. But it isn’t the case that esoteric knowledge in the ancient world was not deliberately kept as a secret. The various priesthoods and sages might have wished that every living person should know what it is, but because esoteric knowledge is often puzzling and counterintuitive in nature, it is easy for what it is to be misunderstood, and then reconstituted as something else. Knowledge which is misunderstood is worse than no knowledge at all. 

I think that is the general pattern of the loss of understanding. Some insight into the nature of reality itself is achieved by an individual, and that insight is taught. An entire culture is developed on the bedrock of that insight. Then it is taught badly, or taught to those who cannot understand what they are being told. The insight becomes something other than what it actually is, and then serves some other cultural or social function. Once a culture is on that slope, it is hard to put the slide into reverse.

I can give some examples here, plus pointers to articles in which I’ve discussed the disconnects between ancient and modern understanding.

First, the ontological argument, which constitutes a huge obstacle to an understanding of ancient thought about the divine. The principal problem with the modern ontological argument is that, for the past seven or eight centuries or so, the argument has been formulated on the basis of the notion that the property of ‘existence’ indicates that something is real. Which is why the argument attempts to provide credible grounds for saying that God exists, and is therefore real. Which is tantamount to a denial of transcendent reality, or the transcendent nature of god. There are other issues with the argument.

I discussed both Anselm’s and Descartes’ versions in The Sacred History of Being, and an extract from the article on Descartes is at: The Ontological Argument In Descartes.  

How did the Christian church, and the philosophers, make this mistake? Probably because they were afraid of the unbounded, the limitless, and the concept of infinity. They wanted an omniscient and perfect entity as their god, but not one that was necessarily beyond human fathoming, and wrapped around in puzzles and paradoxes. They were making God in their own image, rather than the other way around.

I’ve written about the history of the idea of infinity in mathematics, religion and philosophy, via a critical review of Adrian W Moore’s BBC radio series on the subject, broadcast in 2016. My review identifies what is missing from Moore’s tendentious argument.  A long read, but worth the time, I think. It shows just how much pain and intellectual difficulty the idea of the infinite can cause the human mind: ‘Evading the Infinite’

They still had the notion that God lay somewhere behind the physical world, but that notion was not a very sophisticated one. They didn’t need a sophisticated version of God any more, since the main function of religion in the West, from sometime before the fall of Rome, was to support temporal power. 

Divines in the early modern period ended up splitting into two major factions. The Theists saw the world as the creation of God, and that God could be petitioned by Man. Though they had by this time lost all understanding of the mechanism which might make this connection possible. The other major faction was the Deists, who conceived that God had created the world, but that God was the prime mover, and once the world and the cosmos was set in motion, he did not interfere. This view took hold particularly after Descartes and Newton, and the regularity of the motions of the heavenly bodies was then taken to be evidence of the existence of God (this view became known as uniformitarianism, meaning that the forces which were in play in the past, are the same ones which are at play now. The implication is that everything can be calculated).

Much of this is down to early modern misreadings of Aristotle from the 12^th century CE onwards, and his notion of ‘substance’ (ousia). When is Aristotle talking about matter, or about Being, and/or Reality itself? In the Middle Ages they weren’t very sure. The earlier Arian controversy in the 4^th century CE also revolved around substance (is God one or Triune? They decided that God was Triune – ‘Father, Son and Holy Ghost’ - but the component parts ‘were of the same substance.’ (homousia) What that common substance actually was, was never established in Christian doctrine (and the trinitarian view of God is still not accepted by all Christians). The triadic nature of the divine in Christianity is something which has its origins in Egyptian religion. It’s a feature of Christian theology because the ‘Father, Son and Holy Ghost’ are mentioned in the Old Testament. Just without the political fudge of the homousia, which is the legacy of the Council of Nicaea, held in 325 CE.

   

Reliance on Aristotle as a guide always was an error of judgement among scholars. He sometimes argues something other than what he actually thinks (he wrote a book on rhetoric, the Sophistical Refutations), and clearly sometimes actively intended to mislead his students. A case in point is his Nicomachean Ethics, the argument of which proposes that the Gods, by their nature are wholly contemplative, and cannot engage in the world.  Which is an odd and essentially heretical kind of argument to make in the late 4^th Century BCE. The book is a series of lecture notes, probably by different hands (there is an obvious duplication in the text in Book 8). We know that Aristotle held different classes in the morning, and in the afternoon. In the morning he taught exoteric knowledge, in the afternoon he taught esoteric knowledge. The notes on ethics must come from his exoteric teaching, since the argument arrives at, what was at the time, a nonsensical conclusion.

Why would Aristotle do this? The point of spending time building a painstaking argument that leads to a nonsensical conclusion is to find out who the intelligent students are in your class: the intelligent ones are those who pick at the inconsistencies and absurdities of your argument. Those are the ones who you allow to move on to study esoteric thought. Something like this procedure is still used by tribal elders in Africa: you ask a question about the gods, and they give you what sounds like a plausible reply. But it isn’t the real one. The student will only get the real reply once he has shown he is no fool. And that takes a lot more questions on the subject.

Aristotle’s greatest and most dangerous legacy to us also comes ultimately from his exoteric teachings, and is a main cause of our intellectual exile from antiquity. These are his Laws of thought. These Laws became the basis of formal logic as we understand it in the modern world, and they are also the basis of much theological argumentation from the middle ages onwards. Aristotle did not write a treatise on the laws of thought – the various component parts are mentioned and described in his Metaphysics. Put in the most simple way, The first is the Law of Identity (a thing is itself, and not something else), the second is the Law of Non-Contradiction (a thing cannot be one thing and another thing at the same time); the third is the Law of the Excluded Middle (a thing cannot be halfway between what it is and something else).

You can see that the laws revolve around the idea of identity. But it is a purely secular conception of identity, not at all applicable to something which is transcendent.  If you accept the universality of these laws, you cannot discuss the transcendent in any meaningful sense, even if you have academic tenure in a theological seminary. And it is entirely impossible to understand anything useful about ancient religion and ritual, except in purely phenomenological terms.

I have pointed out however that an alternative logical modality has come down to us in the pages of Plato (in the Timaeus), and it is this one which illuminates ancient religion and philosophy. The Neoplatonist writers referenced it. The account of this modality in Plato is rarely discussed by specialists, in terms of what it is and what it means, though it is often read, since Plato’s Timaeus is a text often set for students. Detail available in: Logical Modality in Classical Athens. You might find it interesting then to read The Greek Ontological Model in the 1st Millennium B.C.E.

Though the idea of Being was discussed in classical Greece, by Parmenides and Plato, as well as the Presocratics, the idea was not much discussed in late antiquity, except by the Neoplatonists from the third century CE onwards (Proclus, Iamblichus, Porphyry, Olympiodorus and Plotinus, and some others). But very few outside of their circles were paying attention at the time. Their moment came as late as the Italian Renaissance. And then they were lost sight of once again after the Newtonian revolution.

So the ideas of the Divine as coterminous with Reality itself, and that Reality itself was entirely transcendent in nature, dropped out of philosophical discourse within the Church. It was not picked up again by philosophers in the early modern period. Hence the endless and vacuous discussions about whether or not God could be said to have the property of existence.

With the advent of the European Enlightenment, the history of human thought began to be written about again in detail, for the first time since Diogenes Laertius wrote his Lives of the Philosophers. But the rewriting was done on the basis of certain presumptions. It was axiomatic that abstract ideas,  such as Being,  were difficult things to discuss, and it was agreed that the Greeks had been the first to do this, during the middle of the first millennium BCE. Why was this decided? The evidence for the Greeks was unquestionably in the surviving texts. Whereas it was decided that there was no evidence that other cultures had ever risen to such levels of discussion.

At the time, it could be argued, it would have been difficult to have come to any other conclusion. Several ancient writers however, had suggested that philosophy was an immensely old discipline, including Plato and Clement of Alexandria. Clement even gave a list of the cultures who practised the discipline. Both Diogenes Laertius and Aristotle suggested that it might have had an origin elsewhere. Aristotle suggested that it required a relatively leisured professional class to emerge, by which he probably had the Egyptians in mind. Isocrates (a rival to Plato, and actively opposed to Plato’s outlook) mentioned the Egyptians in connection with philosophy four times. These statements were (and still are) dismissed.

The European Enlightenment marks the point where the Egyptians ancient reputation as philosophers was systematically downgraded by German scholars (documented in detail in Martin Bernal’s Black Athena). In the end it was agreed that the Egyptians dealt in terms of concrete ideas (whatever that means), and consequently were not capable of abstract thought, and were not capable of contemplating ideas of a wholly transcendent level of reality. Even now, Egyptologists refer to ancient Egyptians as ‘pragmatic’ (as opposed to philosophical) in the way they thought about their religion and rituals.

All this was done in order to preserve the Enlightenment fiction of a stupendous achievement of the Greeks in the middle of the 1^st Millennium BCE, by isolating and downgrading contemporary cultures which might have influenced their development.

Over the past 170 years, an enormous amount of evidence has come out of the ground, and quite a lot of it suggests (to those who are awake at least) that philosophy, transcendental thought, and a sophisticated concept of Being was at the root of a number of cultures around the Mediterranean and the Near East. However Egyptologists (in particular) just ignore this evidence as irrelevant to their profession. Most classicists do too, but with some of them, the dam is starting to break.

That’s the short answer to your question…. There is much more, however, which I will skip for the moment. Where are we going on this long trajectory? I had an exchange of mails with a friend in San Francisco in 2015, where we discussed this question.  I took out personal stuff from the exchange, anonymised her, and turned it into an article which is on my website. I will point you at that.  Locke, Newton, and the Rejection of Reality.

I entirely agree with you that the materialistic paradigm necessarily results in the denial of the possible non-locality of consciousness. If you think that thought and consciousness itself are the products of biological processes in the brain, the idea that consciousness is not localised in the human head just sounds like so much nonsense.  The non-locality of consciousness has been a subject for a few philosophers over the years, but is not discussed in terms of the paradigm which we clearly have in common. P. F. Strawson was one, Wittgenstein was another (Strawson was influenced by Wittgenstein’s thought). Among philosophers it is generally referenced as the ‘no-ownership’ model of consciousness. Usually the argument is along the lines of: “we are constricted to think as we are about identity, perception, experience, physicality, location, etc., by the limitations of (apparently) being physical entities in space and time. Which may not actually be the case”. Not much fun to read, but at least they understand that a materialist interpretation of human existence is problematic.

I also agree that there is a lot of evidence which suggests that consciousness is not localised in the brain, or associated with matter at all. Consciousness may interact with matter, and have some kind of interface with the material world, but that is not the same as being a phenomenon which is entirely localisable. There is a two-way traffic, as you said earlier in your mail.

For me I understand this to be possible because things which are apparently in different places, are actually together in one place, which I reference as the plenum. Which is a place which is not localisable. It is also the key idea we have lost, which makes it impossible to understand both reality itself, and the representations of it, in which we live.

The implication of the concept of reality being a plenum (as it was understood in antiquity) is that both God and gods are necessarily real. This is the consequence of the way they were defined. Each of them has an excellence special to them. That excellence is what makes them real. The plenum is itself real because it is complete and perfect in itself [...]. This fundamental idea is what underpins Greek and Assyrian conceptions of the importance of excellence in thought and activity. It is also why excellence in craft skills was venerated in antiquity. Excellence is what connects everything with everything else. It is why the perfect  execution and observance of a ritual or ceremony was regarded as the most important thing to achieve. This pattern of ideas served as a model among the Greeks for the human soul, and to the organization of society, discussed at: Justice and the Harmony of the Soul. Excellence was also what gave access to the divine: Excellence and the Knowledge of Divine Things.

So I accept that consciousness can be understood as a bridge between ourselves as we are in the form of finite representations, and what we are in that place (which is paradoxically no place) which is filled with the complete and the perfect.  And I agree with you that an important aspect of ritual is about facilitating traffic between the worlds. As you suggest, the traffic is a bit constricted now, since many fewer people have any conception of this purpose…. However, it is necessary for ritual to be performed, and observance to be made, but not essential for the process to be fully understood for it to have an effect. There are still people performing rituals and observances.

   

I read somewhere (in what I’ve read of your work so far), that myth is often about travelling to that other place, either by ascending to Heaven, or descending to an underworld (actually the same place, the plenum, just dressed differently). I agree. [...] The creation of a myth, and the telling of it, can be understood as performative also, particularly if rituals are associated with it: it is ritual made visible. Placing myths in the sky, to descend into the earth, and to rise from it once again, as part of a regular cycle, like a rotating Tibetan prayer wheel, is performative and served the same function.  

Best, Thomas

Pharaoh Akhenaten, the Aten, and the History of Ideas (I)

[An extract from recent correspondence with a specialist in ancient astronomy. Akhenaten's reign is not often considered to have been one concerned with a technical understanding of astronomy (though through his veneration of the Divine Aten, his interest in the sky is clear). There are interesting reasons for this neglect]

January 24, 2019

[...]

I’ll respond to your mail in order. First the Egyptian paper.

I had the opportunity to look at pharaoh Akhenaten in some depth at university. His reign was covered in the general course on Egyptian history, largely from the point of view that most scholarly writing on Akhenaten doesn’t fit very well with the evidence. I liked this aspect of the teaching of ancient history in London: a highly critical approach to what could actually be known, which sometimes isn’t very much. Other things I disliked, but I’ll get on to that later.

The title of the upcoming paper is ‘The Horizon of the Aten’ – the Aten being the sun disk, which is most associated with Akhenaten, though not exclusively. Most writing on Akhenaten suggests that his religious reforms were actually innovations, which the priesthoods – particularly the priests of the cult of Amun, probably the most important divine cult in Egypt in the 14th cent B.C.E – regarded as heretical.

At the least, the religious changes instituted by Akhenaten, threatened the power of the cult. It does not follow however, that such changes represented innovation and heresy, whatever the priests of Amun said.

After a little digging last year, I came across a piece of information, never mentioned in college, and which did not appear in any of the books on the reading list. It surfaced first in a book published relatively recently. It is evidence which appears in contemporary texts. It wasn’t deemed worthy of a mention in any of the many studies of the 18th dynasty, because it didn’t appear to make any significant sense in that context.

For me, it was like a bolt from the blue. It made sense of half a dozen other pieces of information rattling around in my head, almost simultaneously. Akhenaten now looked like the adherent of one of the oldest cults in ancient Egypt. This ‘new’ information, made sense of what he understood himself to be doing (it is hard to understand any ruler who builds a temple with 400 sacrificial tables, as he did). A reformer, but not an innovator or heretic at all. The evidence ties his reforms, and the concept of the Horizon of the Aten, with the equinoxes and the precession. A reform which was intended to restore a pattern of thought which was contemporaneous with the earliest of Egyptian dynasties. Several features of the Giza plateau would not otherwise be present. They set things up so they could not possibly have missed the precession.

The new paper connects texts, architecture and cultic evidence to make the case.

Thanks for the information about the King Den tablet, which I had not seen before. I think you may be right about the astronomical aspect to the image. One of the problems in studying ancient history, particularly in the ancient Near East, is that real events are often presented in terms of mythical images, since what happens on earth echoes what is in the sky. The sky is the more real. So sometimes it is difficult to tell whether an image represents a myth, or whether it represents a physical event dressed up in mythical imagery.

A similar issue of what is being represented occurs with images of the gods – when are representations of the gods just that, and when are they the gods themselves? I remember attending a lecture in Cambridge more than twenty years ago where the speaker spent twenty minutes talking about this question, and why the Egyptians would create a statue of a god standing on a sled (they did), which indicated that it was a representation of a statue of the god, and not a representation of the god itself.

I first read Hamlet’s Mill sometime around 1978, which is a long time ago now. I accept the argument, but the book is in many ways poorly constructed, though there is a mass of useful information in there. The fugal aspect of the structure was not likely to be appreciated in academia, and the text is very dense, so I think they did not give a lot of thought to who was going to read it (sometimes writing a book for yourself is the only way). I have read academic books which are much more difficult to read, but which have credibility among academics – I’m thinking particularly of Lynn Thorndikes’ eight volume ‘A history of magic and experimental science’. No points for style or clarity.

I’m not a mathematician, but I’m comfortable with numbers. I worked out pretty early on that the rate of precessional movement means that the intersection of the celestial equator and the ecliptic moves a degree in about 70 years. That is about two diameters of the moon as seen from the Earth. How blind do you have to be in order not to see that? Especially if those observing the sky belong to long-established professional priesthoods, with a background in astronomical observation and lore, and in possession of a temple library?

Why was Hamlet’s Mill shunned by academics? Because as far as the scholarly consensus goes, the argument of the book is faulty in conception, as well as execution. The book was not simply ahead of its time, but it argued outside the paradigmatic frame in which scholars were willing to consider evidence. That’s an important thing in academia: the field in which scholars operate has to have its limits defined. They study and write within a frame in which it is possible to make a case for this or that. It is a mistake to think that academics engage with what the whole range of evidence might have to tell us. It is not about understanding, but about developing an understanding of the subject under study.

This is not altogether a bad thing. Sometimes it is necessary to narrow a focus of interest in order to understand what is being considered. I’ve often engaged in this kind of process myself. It becomes problematic where the artificially constricted parameters ossify into the definition of an academic discipline or sub-discipline. When people talk about the value of interdisciplinary research, they actually mean that the nature of much contemporary research is too narrow to keep students supplied with interesting or useful Ph.D proposals…

Which is why I am working outside academia, and self-publishing my work.

Yes there is a reluctance to see the prevailing paradigm as untenable. But you are right that established paradigms are always difficult to overturn, just because they are established, and a lot of scholars have an investment in preserving the status quo. Is there another agenda? Yes there is. In fact, there are several. For the moment I’ll just pick out a couple of them here (we’ll get through all of them in the course of our conversation I’m sure!).

1} Not everybody who writes about human intellectual history has a practical grasp of astronomy. Or philosophy. Or theology. And sometimes their heads are filled with nonsense ideas, many of which first saw the light of day as late as the European Enlightenment. One of these nonsense ideas is that our distant ancestors were subject to a primitive stupidity, for which the Germans have an appropriately ugly word: ‘Urdummheit’. It is the myth of progress over time. If that notion is firmly
planted in a scholar’s head, then it is clearly straining the bounds of possibility to credit ancient peoples with any kind of understanding of precession. It is just common sense that they wouldn’t notice such a thing. Primitive stupidity is a scholarly presumption which is powerful enough to override the need to engage with facts and observation. There is less of this presumption in academia than there used to be, but it is still there, and not always hiding in the corners.

2) Historians in academia generally work within a paradigm which owes its character to the writings of Karl Marx, as mediated through the writings of Max Weber and other sociologists. This way of looking at things got its foothold in Britain through the London School of Economics in the early years of the twentieth century. What it means is that society is studied in material and economic terms. Ideas are deprecated as fundamentally irrelevant to the forces which have shaped history and human cultural development. This appreciation of the forces which underpin our society is assumed to be universally true, and so is equally applicable to the study of ancient societies. If ancient scholars did not know this to be the case, it was because they were victims of what Marx and Engels called ‘false consciousness’. In other words, history was to be explained in terms of power and propaganda, and ideological struggle between competing groups. All my teachers at university shared this point of view, to some extent at least, but some did allow that it might be too narrow a perspective.

From this point of view, if you are talking about priestly ideas, which you are if you are talking about ancient astronomy, you are talking about a subject which is irrelevant to our understanding of antiquity. The real history of humanity is to be teased out of a knowledge of priestly power and the propaganda developed to support that power. Religious ideas and iconography serve that priestly power, and are to be interpreted in such terms. So the detail of a concern with astronomy, and a knowledge of precession, is, for some historians, beside the point.

I describe myself as a historian of ideas. It is a scholarly discipline, working outside the Marxist-inspired model, which came into existence in the mid 1930s, specifically to deal with the materials and questions other historians were ignoring. The book which kicked-started the discipline was authored by Arthur Lovejoy. It was called ‘The Great Chain of Being’. It considered the concept of ‘plenitude’ from the Greeks up to the European Renaissance. My first book can be understood as an extension of the argument into more remote times.

Best, Thomas

Orientalism and the Post-Enlightenment Agenda (Writing to Katherine Blouin)

[In 2018 I had some brief conversation with Dr. Katherine Blouin via Twitter. Which prompted the following email]. 

 I thought I should tell you about my work, since I am part of the largely disorganised cohort working for a radical change in our understanding of the role of ancient Greece in the development of western civilization.

My initial background is in classics and philosophy. At UCL I also studied Mesopotamia (languages and culture). This study was across three departments - UCL ancient history (text corpus, with Amelie Kuhrt), UCL department of Hebrew (Sumerian, with Mark Geller), and SOAS for Akkadian language (mainly with Andrew George). As a result I left college with a view of the relationship between Greece and Mesopotamia which was quite different from what I was led to believe while I was there.

 I graduated in 1992. I've been writing on the relationship between Mesopotamia and Greece since then. I arrived late to Said's work, since I was initially in thrall to Martin Bernal's thesis in Black Athena (though I did attend Said's series of lectures at UCL in the fall of 1993).

My main interest is in the history of philosophy. The conventional history of philosophy is pretty much a fabrication, The evidence doesn't support what is said about it, and it isn't actually that difficult to detect the fraud. Which led me to consider why the fraud was easily accepted. The answer turned out to be, as Bernal suggested, the post-Enlightenment agenda. The Enlightenment sought to understand everything in terms of common sense explanations, and not much in ancient philosophy conformed to common sense ideas.

So I've been researching and writing, off and on since 1993, about the origins of philosophy, Greek philosophical writing, and the cultural history of the west.

I was also interested in the role of anthropology in our understanding of past modes of thought. In 1993 I wrote a study of Frazer's essay on Plato, in which he argued that the idea of 'Being' was a barren notion, and nothing useful could be said of it. That is not what they thought in antiquity. So his later work was essentially a reframing of the past with one of its most important concepts entirely written out (formally published in 2016).

Hence I wrote The Sacred History of Being, which was finally published in November 2015. The point of this book was to show two things (amongst others): that the concept of Being was of immense importance before Plato, and that modern arguments about the nature of the Divine did not at all resemble ancient arguments, and so we could not easily understand what the ancients were talking about.

So we have a real problem. I've published a couple of volumes since The Sacred History of Being, which largely serve to illuminate how the ancients understood both physical reality and the Divine. Without a grasp of their radically different way of understanding, much of the ancient world is just unintelligible (oracles, omens, sacrifice, extispicy, etc).

These extra volumes serve as the basis for understanding that most ancient divine cult was transcendentalist in nature. Meaning that religious thought already embraced ideas and abstractions which surface in the writings of Plato and others. In Mesopotamia, in Israel, in Egypt, in Phoenicia, etc.

Which brings me to where I am now. I showed in The Sacred History of Being that the Assyrians and Babylonians were capable of abstract thought, and conceived of the Divine as a pure abstraction. They understood the basis of dialectical thought (collection and division) which the Greeks are credited with. They embraced the idea of excellence as the Greeks did, and the concept informed the education of their kings. They understood morality to emerge from the logic of excellence, as the Greeks did. They also conducted public debates on the relative merits of this or that good thing, as the Greeks also did.

 I intend next year to publish a concluding volume in the series which provides the evidence for these Assyrian and Babylonian parallels.

 After that, the jig will be up for the classicists. But I have no expectation that they will take notice of what has happened.

 One of the [key] things they have taken on board from their subject is the pursuit of excellence. They pursue this through what they understand as the rigourousness of the study of their source materials. The context of these materials, and the rest of the world, have relatively no importance for them. So why should they change their views?

Thomas Yaeger, October 18, 2018. [Reply received]

Reviewer Notes for 'The Sacred History of Being' (2015)

Notes on Part One:

The Preface gives an outline overview of the book and its parts, and something of the context. One way to look at the book and its subject is as an extension of A.O. Lovejoy’s The Great Chain of Being, which examined the idea of Plenitude from the Classical Greeks onwards. SHB provides an extra nine hundred years to the history of the idea of Being, taking it back into the 14^th century B.C.E.

The first chapter explains something of the authors background, and how he came to approach the past from the point of view of the history of the idea of Being, and came to study ancient languages and history in London.

Historians of philosophy treat the 5^th century B.C.E. as the proper start of sophisticated philosophical thought, as we now understand it, with figures such as Parmenides, Plato, Socrates, and Aristotle as the principal figures. It comes as shock to find Plato recounting the immense age of philosophy in his dialogue The Protagoras. This is the subject of ‘How Old is Philosophy?’, and ‘The Arrival of the Idea of Being’

‘The West and the Other’ explores the way western modes of thought are inverted from those in the east.  ‘The Golem’ is about how it is possible for things present in bodies of evidence to remain invisible. When the SHB project began, I expected it to be difficult to find supporting evidence. In fact it is present in vast quantities both in ancient texts and in the archaeology. But two centuries of scholarship have glided past this evidence, as if it is not there.

‘Change and what is Permanent’ looks at some of the ideas which are present, but which are not much studied. These are abstractions, which are normally associated with the philosophical explosion of the 5^th century B.C.E. in Greece, but are major components in thought from earlier periods. The human capacity to deal in abstractions is, as a corollary of the notion of cultural progress, assumed to have developed first in Greece. This a poor frame in which to try to understand ancient thought.

The discussion which occupies the rest of Part One of SHB concerns the Ontological Argument, which has been the mainstay of the understanding of the nature and reality of God in the west, since Anselm. The point of this extended discussion of this argument is to show that it isn’t really about the nature of the divine at all, but about the properties and attributes of the divine. In other words, it is loosely argued, and assumes that the frame of time, space and the underlying reality of the world (whatever that might be), is something which exists apart from the nature of God.

This isn’t how divinity was understood or discussed in the ancient world. To argue in this way puts an insuperable barrier between ourselves and the possibility of an understanding of the divine in antiquity, and makes it impossible for us to understand the relationship between religion and philosophy.

The chapters are:

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.
The Ontological Argument.
The Ontological Argument in Anselm.
The Ontological Argument in Descartes.
The Nature of Reality in Berkeley.
Hume and Kant on Reality.
The End of the Ontological Argument.

Notes on Part Two:

Part two explores ancient Greek ideas about the Divine, and the arguments which were used. It looks in detail at argumentation which appears in Plato’s work, and also at ideas which are reputed to have been part of the doctrines of Pythagoras.

‘The Sweet Song of Swans’ examines the nature and history of the scholarship around the work of Plato, which is contained mainly within the period of the past two centuries. The Platonic corpus is problematic. It appears to be inconsistent. Plato was still writing at 80 years of age, and so it seems possible that Plato changed his views over time, and this is reflected in half a lifetime of writing. On the other hand, there is no evidence that Plato’s Academy was a research institution, and so there is no reason to assume that his views changed during that half a lifetime. Those modern scholars who take this view have a problem however, since they have been unable to dig out the consistent doctrine (hoary with age) underpinning the weaving discourse of the dialogues.

The late neoPlatonic philosopher Olympiodorus gives us some clues as to the reasons for the form of Plato’s discourse. Further clues come from close analysis of some parts of Plato’s dialogues, particularly in the Timaeus and the Sophist.

Plato often refers to looking to the ‘one true thing’. That ’one true thing' is Being itself, which he characterises as ‘The Good’. It never changes, it is entirely itself, and it is the goal of the philosopher to rise through the forms to the form of the good, and to return again, with knowledge of beneficial things.

J.G. Frazer once said that nothing useful could be said of the idea of Being. This was flying in the face of the whole of antiquity, whose civilizations articulated many aspects of the nature of Being as part of their understanding of the world. ‘Eleven attributes of Being’ discusses these, and why they had such importance in antiquity.

‘Pythagoras on Totality’ concerns a doctrine which was understood both in Mesopotamia and in ancient Greece. The idea can be traced back to the middle of the second millennium in Mesopotamia. Pythagoras is said to have learned of it in a lecture at Babylon after its fall to the Persians. Plato knew of it, and mentions the doctrine in his Timaeus. Once again, it is the idea of the ‘all’ being the location of all knowledge. It is the principal goal of the philosopher to access the totality of all reality. The distinction we customarily imagine to exist between philosophers and priests in antiquity, more or less breaks down at this point.

‘Solon in the court of Croesus’ is a re-examination of one of the most famous stories from antiquity, as told by Herodotus. There are details in the story which do not make sense without some analysis. In fact he is telling us about the importance of completion in understanding whether something is good or bad. He also connects the life of man with the cosmos using an unusual selection of numbers.
 

If we look closely at Homer’s Iliad (Book Eighteen in particular), we can see ideas in the text which are later reflected in Plato’s writing, which suggests strongly that Plato was writing about a body of ideas which was present in the late Bronze Age (discussed in the chapter 'Being in Homer').

The chapters are:

The Sweet Song of Swans.
The Academy.
The Platonic Theory of Being.
Plato’s Theory of Vision.
The Paradox of Knowledge.
Eleven attributes of Being.
Pythagoras and Totality.
Solon in the court of Croesus.
The Complexion of the Dead.
Being in Homer.

Notes on Part Three:

Ideas about water and ocean are held in common around the Mediterranean in the first and second millennia B.C.E. These ideas are associated with creation and the generation of life. They are also associated with the idea of abundance, which features in the Babylonian creation myth, performed each year by the King and the priests. The liturgy tells us about the intellectual frame of their world, as they understood it.  The physical world is a place of refuge, created by the gods to ensure the well-being of human life. The King is the earthly embodiment of the King of the gods, Marduk, whose character is described in detail. Each of his qualities enables him to maintain the good order of the world. Which tells us important things about how the gods were understood.

Israel suffered from the Assyrian addiction to war and cultural depredation. The Hebrews also spent some time in exile in Babylon, which gave them an insight into Mesopotamian thought about the Divine. Or was that insight garbled by the Hebrews, because they were engaged in a political and military struggle with their neighbours? ‘The Idea of Being in Israel’ considers this question.

The close connection between Hebrew and Assyrian thought was demonstrated by Simo Parpola in the early nineteen-nineties,  who showed that the Jewish Kabbalah is a close analogue of the Assyrian sacred tree. The Assyrian sacred tree is never described in detail in texts, though it is ubiquitous in Assyrian iconography. However if the principal Assyrian gods are placed within the structure of the Kabbalistic tree, with their associated god numbers, the fit is perfect. Formerly the Kabbalistic tree was understood to have been devised sometime in the early middle ages, possibly drawing on gnostic thought. If it is as old as the Assyrian sacred tree, then the history of intellectual thought as we have it is wrong.

The final part of the book concerns the installation of statues of the gods in both Assyria and Babylon. The revelation of close examination of the texts is that they are not talking about installing images of the gods, but the gods themselves. The rituals are described in detail, which gives an extraordinary insight into the minds of the scholars in the royal courts of Assyria and Babylon, and suggests we need to alter our perceptions of how divinity and the gods were understood in the first and second millennia B.C.E.

The postscript summarises how we might now understand the concept of Being, and the meaning of the representation of the gods, in both ancient Mesopotamia, and in ancient Greece.

The chapters are:

Ocean and the Limit of Existence.
Creation.
The Fifty names of Marduk.
The Idea of Being in Israel.
Understanding Creation as a Sacred Tree.
Being, Kabbalah, and the Assyrian Sacred Tree.
The Making and the Renewal of the Gods.
The Ritual sequence and its purpose.
The Nineveh ritual.
The Babylonian ritual.
Finding the Name of the Sacred Tree.
Postscript.

Notes on the Appendices:

It is a strange fact that neither of the two premier universities in England in the late eighteenth and early nineteenth centuries, managed to get the writings of Plato, Aristotle, and the neoPlatonists into English translation before a private individual did, at the cusp of the two centuries. The work was done by a scholar of great gifts, Thomas Taylor, supported by a private benefactor. He chose to undertake the translation of Plato and Aristotle, and the neoPlatonists, because he did not recognise the notional division between them which has taken root in western academia since the Enlightenment. Post-Enlightenment scholars like to understand Plato outside of a theological context, which doesn’t at all square with the focus of his writing. Plato would have been utterly dismayed to be read in this way, but the neoPlatonists are clearly talking theology, and so they need to be kept apart for the purposes of an academic and untheological understanding of Plato. I’ve included a mildly redacted piece by Thomas Taylor, which explains why Plato’s discussion of the one and the ineffable is a theological matter of some importance.

The appendix on the story of the first sage in Babylonia is included to amplify what we know of the Babylonian understanding of how knowledge is acquired. It is acquired from Being itself, represented by the ocean and the ocean depths, because Being is the all, and all knowledge is necessarily already there. So a conversation with eternity, which is what the all is, is the principle source of knowledge of good and beneficial things.

The extract from the annals of the Assyrian King Ashurbanipal on the exercise of kingship shows the importance attached to the idea of excellence in ancient Assyria, and that a concern with excellence was not an invention of the Greeks. Ashurbanipal lived in the seventh century B.C.E. Before now this significant passage had not been retranslated since the late nineteenth century. The Assyriologist Simo Parpola was kind enough to provide a new translation for me in 2005.

The appendices are:

Thomas Taylor on the Ineffable principle.
Oannes and the Instruction of Mankind.
Ashurbanipal on the exercise of Kingship.
Select Bibliography.
Abbreviations



Thomas Yaeger, April 10, 2018. Chapters added, September 2018.

Philosophical Thought in the Neolithic (and why we cannot see it)

While responding to a mail from the managing editor of a scholarly magazine,  who I'd pointed in the direction of a couple of articles on my blog, I found myself condensing my project into the compass of just four paragraphs. This covers both what I am doing, and why. I reproduce them here:

Years ago I read Adolf Erman's collection of Egyptian Literature in English translation. One text in particular made me sit up, since it involved the use of concepts I associated with classical Greece. The text was Akhenaten's 'Hymn to the Aten'. Akhenaten lived some nine hundred years before Plato and Aristotle, yet he seemed to be comfortable with concepts which are supposed to have been first discussed among the Greeks. I looked for other 'anticipations', and found them.

That propelled me to study in London (in 1989). By that time I'd started to look at antiquity quite differently from the way I'd been taught by classicists. Instead of seeing classical Greece (and the work of the later philosophers all the way up to the closure of the philosophical schools in 529 CE) as the beginnings of sophisticated thought about abstract concepts, I began to treat major components of classical thought as the possible end point of a way of thinking, which might conceivably stretch back into the neolithic. The notion of excellence is one of those components, which as we now know was a concern for the megalith builders.

That's the basic hypothesis. The idea of Greece as the birthplace of philosophy is a modern construct. Plato himself said it was a very old discipline, but classicists don't bother to discuss what he had to say about that when they are writing about the history of philosophy. The evidence from around the eastern Mediterranean however, bears out what Plato said.

My focus so far has been mainly on Mesopotamia, and back to the 14th century BCE, because it is the area where I have the most expertise. It is now pretty clear however that the hypothesis that the presence of abstract ideas stretches back into the Neolithic is sound, and that this insight can deliver great riches. If only we can escape our enlightenment presumptions about the intellectual poverty of our distant ancestors.

Best regards,

Thomas Yaeger

The two articles the managing editor was pointed at were: Stone Circles, Phenomenology, and the Neolithic Mind, and: Frazer and the Association of Ideas, 

I introduced the first article as looking at the context and function of Stone Circles, particularly when looked at as structures which may have served a similar function to divine statues in both Greece and Mesopotamia. In Mesopotamia, during the installation of divine statues, the three day ritual necessarily involved pointing them at defined areas of the sky, because the heavens were understood to represent an image of the Divine - of Being and Reality itself. 

The second article I described as a review of J. G. Frazer's approach to the meaning of evidence from the ancient world. He was trained in Classics, and knew the works of Plato virtually backwards and his concern with the idea of Being, but chose to write his account of ancient thought entirely without reference to the idea of Being in antiquity. He suggested that since nothing could be predicated of Being, it was an entirely barren concept. So Frazer's voluminous output is really a species of fiction, replacing sophisticated thought about the nature of reality with the argument that ancient human thought was, more or less entirely, built on intellectual error.

Evading the Infinite: a Review of Adrian Moore’s ‘A History of the Infinite’

[This chapter, 'Evading the Infinite',  is one of twenty-one essays in the book Man and the Divine, published in August 2018. The book is available in ePub format from leading retailers of eBooks, such as Barnes & Noble, Blio, Kobo, Itunes, Inktera, Smashwords, etc. Information about Man and the Divine can be found here]

***

This is 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. Episodes are discussed in order of broadcast. I was in conversation with Adrian Moore from halfway through the series, and this conversation continued into early 2018.  Moore wrote a book on the philosophical and mathematical aspects of the infinite in 1990, which was a volume in the series ‘Problems of Philosophy’. This radio series gave its arguments much more exposure than they ever had before.

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. 

Moore tells us that Plotinus’ idea that the divine was infinity itself was taken up by Thomas Aquinas, Aquinas was schooled in Aristotle (there was a renaissance in the study of Aristotle at this time, which was not mentioned), and he was aware of the distinction which Aristotle had made between a potential infinite, and the actual infinite. He also became an authority in the Catholic Church, and wanted to incorporate the philosophy of Aristotle into Christian thought, but with an emphasis on the idea that the divine had reality in the world in terms of a potential infinite, rather than as actual infinity. Aquinas essentially divided infinity into a mathematical conception, and a metaphysical or theological conception.

Aquinas’ ideas went on to dominate later Christian thought, but not everyone agreed with what became the doctrinal position. Much later again (Moore’s argument makes large temporal jumps), the arrest of the scholar Giordano Bruno in the late sixteenth century, showed how dangerous the idea of the Infinite could be. He was imprisoned and tortured by the Inquisition for having argued that the universe was infinite, and that there existed an infinite number of worlds. He was burned at the stake in 1600. Moore suggests that this marked the end of the renaissance.  Galileo Galilei was lucky the same thing did not happen to him. His interest in the number of numbers in existence was discussed in this episode, and these and the numbers of their squares and roots, seemed to him to be infinite. This represented the birth of a whole new set of paradoxes with later repercussions in mathematics.

The fourth programme 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.

In the episode which follows, the subject is the nature and development of the calculus. It begins with the observation that to divide zero by zero, or zero into anything at all, makes no sense. If you know anything about the calculus, it is clear what is being talked about in this episode, but the way it is discussed is lacking in the kind of precise description you might expect. 

A train is used as an illustration. Travelling a distance of sixty miles over an hour means that the train had an average speed of sixty miles an hour. However, the train might have been travelling at a much higher speed for half of the journey, and have been delayed by signal failure during the second half of the journey. So if you measure the distance travelled and the speed at a particular point in the journey, the result may be misleading. If the time period measured is very short, say close to zero, and the distance travelled is close to zero, then you will know nothing useful about how fast the train is going, and how long it will take to complete its journey.  

Calculus enables the accurate measure of quantities which are subject to change (which is why the inventor of the calculus as we know it today, Isaac Newton, referred to it as ‘Fluxions’). The episode makes clear how important the development of the mathematics of change has been ever since, and that much of the modern world depends on the use of calculus. The term ‘integration’ makes no appearance in this episode. 

Much of the rest of the programme discusses the invention of calculus, and the bitter dispute which arose between Isaac Newton and the philosopher Leibniz, who developed a similar approach to the mathematics of change quite independently. Newton appears to have begun to develop the mathematics for ‘fluxions’ early on – perhaps as early as the 1660s. The chronology is not clearly established, but Leibniz may have developed his version some ten years later. 

Newton did not publish any information about the mathematics involved in the calculus until many years later, preferring to share a few details with his friends and colleagues. Newton was aware of Leibniz and his work, not least because he too was a member of the Royal Society. Eventually he wrote to Leibniz with some limited details of the calculus (Moore suggests that Leibniz could not have understood these details since they were in code). Newton became aware that Leibniz had developed similar mathematics to deal with change, and a long dispute ensued, mostly conducted via intermediaries. Leibniz was often travelling, and so correspondence sometimes took months to reach him. Newton launched attacks on the integrity of Leibniz, accusing him of plagiarizing his ideas. Leibniz was bemused by his attacks and the force with which they were made. But Newton had decided that Leibniz was his enemy, and that was that.

Eventually it was proposed that a report be prepared by the Royal Society to establish who had the prior claim to the invention of calculus. This sounds fair, except that the President of the Royal Society wrote the report, and the President was Isaac Newton. As Moore says, ‘not Newton’s finest hour’.

The philosopher George Berkeley makes another more substantial appearance in this episode, since he wrote a criticism of what he called ‘the analysts’ (The Analyst: a Discourse addressed to an Infidel Mathematician (1734)). His criticism was based on the general lack of rigour with which calculus was often used at the time, and argued that scholars who attacked religious and theological arguments for lack of rigour were being similarly careless. The criticism revolves around the limitations of the technique already mentioned, when the quantities and measures chosen are too small to produce intelligible results.

The most famous quotation from this book describes infinitesimals as ‘the ghosts of departed quantities’. The book seems to have been aimed particularly at the mathematician Edmund Halley, who is reported to have described the doctrines of Christianity as ‘incomprehensible’, and the religion itself as an ‘imposture’. Moore references the fact that the technique of the calculus lacked technical rigour until the early nineteenth century, until the idea of the limit was introduced (in fact Cauchy, and later Riemann and Weierstrass redefined both the derivative and the integral using a rigorous definition of the concept of limit. But that is another story). 

Moore concludes the episode by saying that:

precisely what such precision and rigour show, is that the calculus can be framed without any reference to infinitely small quantities. There is certainly no need to divide zero by zero. What then remains is a branch of mathematics, which is regarded by many, in its beauty, depth and power, as one of the greatest ever monuments to mathematical excellence.

The sixth episode 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 eighth episode opens with a discussion of the Andromeda Galaxy, which is the furthest object in the universe which can be seen with the naked eye. Its light takes two million years to reach us. Yet it is a close neighbour to our own galaxy. The distance is mind-numbing, but it isn’t infinite. The central question of this programme is ‘where does the concept of infinity fit in with our attempts to understand physical reality?’
Looking up at the night sky is evocative of the infinite for many of us because of the enormous distances involved, but how appropriate is this? Is the number of stars infinite? Is space infinite? Is anything in nature infinite? The Greeks contemplated these ideas. Moore quotes Archytas of Tarentum (4^th century B.C.E.) on this question:
If I am at the extremity of the heaven of the fixed stars, can I stretch outwards my hand or staff? It is absurd to suppose that I could not; and if I can, what is outside must be either body or space. We may then in the same way get to the outside of that again, and so on; and if there is always a new place to which the staff may be held out, this clearly involves extension without limit. 

Aristotle, who wrote a little later, resisted Archytas’ argument, and said that although it is true that the universe can’t be bounded by anything outside it, nevertheless, it is only spatially finite. This may be difficult for us to grasp, but it has become a staple of contemporary cosmology. Moore asks, how do we arbitrate between the views of Archytas and Aristotle? Moore then quotes the seventeenth century philosopher Thomas Hobbes, who didn’t think there was anything to arbitrate on, since neither view (infinite or finite) is based on anything which presents itself to the mind. So the question (for Hobbes) is meaningless. 

The mathematician John Barrow (Cambridge) distinguishes between speculations about the universe as a whole, and our piece of it, the part which is visible, about which we can have a scientific understanding. So though the universe is potentially infinite in size, we can only see a part of it, which may not be infinite. If someone says that it is infinite, that is a purely philosophical speculation, and cannot be demonstrated. Moore then asks: If we confine ourselves to the part which is visible to us, can we ask if that part of the universe is infinite or finite?

Isaac Newton though that the matter in the universe had to be infinite, otherwise the cosmos would collapse in on itself. The modern consensus, Moore suggests, is that the amount of matter in the universe is finite. And that indeed, space itself is finite.  Even if unbounded, as Aristotle thought. This is an idea which is difficult to grasp – how can the universe be both finite and unbounded? 

It might be possible for the universe to be both finite and unbounded. The example is the surface of the earth. Whichever direction you travel, you will always arrive back at the same place (assuming the oceans are not a barrier of course). There is no limit to the distance you can travel, but the surface of the earth is finite, and not infinite. Space itself may be like this, so that there is no limit to the distance you can travel, but that its extent is finite. 

If space were curved like the surface of the earth, how would that manifest itself? It is possible to distinguish between the properties of triangles on a flat Euclidean surface, and their properties in a curved space. You can tell the difference between these surfaces, because the angles add up differently. It appears the evidence that we have suggests space itself is not curved. The evidence needs to be drawn from examples which involve distances of billions of miles [no explanation here of how this might be done], so that the differences in the angles would show up. 

It is still possible that space is finite, and indeed that distant galaxies are (as they appear to be) moving away from us through the expansion of space. The evidence for this is the phenomenon of the Doppler shift. The same evidence allows us to extrapolate backwards. As the galaxies are moving further and further apart, so we can infer that they were once very close together, and originated in an enormous explosion of energy which may have given rise to the universe as we know it today, a finite length of time in the past.

Moore points out that scientists like to evade the infinite. If an infinite crops up in equations, there is a feeling that a theory is incomplete. Moore suggests that this finite point so long ago, doesn’t eliminate the infinite, it reintroduces it. Mass and energy at the point of the so called ‘big bang’ would have been infinite, if that apparent beginning was a real  event, there would have been infinite density, and infinite temperature and, had the mass not been travelling at great speed, there would have been the danger that the whole universe would have collapsed through gravitational attraction, and there would be a big crunch of infinite density. As may happen in the distant future.With infinitudes, you lose the capacity to predict things. Which is another reason why scientists like to evade the infinite.

Moore concludes by recalling how the series opened, where the attitudes of the Greeks to the infinite were explored. He repeats his view that the Greeks had trouble understanding the infinite, and also found it hard to ignore it. Now we find ourselves in a similar position. Exploration of the cosmos invites us to reckon with the infinite, even when we are not sure what the infinite is. 

The programme following (number nine in the series) began with a scene from the opera ‘The Makropolos Case’ by the Czech composer Janáček. The premise of the opera is simple: more than three hundred years earlier the heroine of the opera, Elina Makropolos, was given an elixir of life by her father, the court physician. She is now nearly three hundred and fifty years old. She has reached a state of utter indifference to everything, and her life has lost its meaning. In the opera excerpt she sings a lament: ‘Dying or living it is all one. It is the same thing. In me my life has come to a standstill. I cannot go on. In the end it is the same. Singing, and silence.’ Makropolos refuses to take the elixir again, and dies.

The opera raises some profound questions, about life, about death, about purpose, and about our finitude. But how should we understand our finitude? Human finitude has many facets. We live in a reality, which for the most part is quite independent of us. We are limited in what we can know, and in what we can do, but importantly, we also have temporal and spatial limits. Though it isn’t entirely clear what those actually are.  Moore asks, as an example, if he began to exist when he was born, or if he was himself when he was still a foetus. Another question concerns how big he is. He gives his dimensions and weight, but points out that you could cut his hair off, or even amputate his legs, without destroying him. Some philosophers would argue that who a person is, is represented principally by the brain of the individual in question.  And other philosophers might argue that we are not physical entities at all.

In any case, it is clear that human beings are not infinite in size.  And, unless there is an afterlife, there will come a time when we no longer exist. Is the prospect of our annihilation something we should fear, deplore, and does it reduce our lives to meaninglessness? The Greek philosopher Epicurus did not believe in an afterlife. But the Epicurians did not fear or deplore death. They did not see how they should be affected by something they would not be around to witness. They were of the view that death was not an evil to us, since we were not around to witness it. Lucretius, also an Epicurean, reinforced the point by saying that we didn’t exist before we were born, and the fact that we won’t exist after we are dead, is just a mirror image of that. Lucretius asked, ‘is there anything terrible there? Anything gloomy? It seems more peaceful than sleep.’

The twentieth century philosopher Bernard Williams went even further. Rather than dwelling on the innocuousness of being dead, he dwelt on the awfulness of being perpetually alive. He wrote a famous article which took both its theme and its title from the Makropolos Case. Its subtitle was ‘reflections on the tedium of immortality.’ He argued that a never-ending life would become what Elina Makropoulous’s life had become – tedious to the point of unendurability. For Williams, it was about whether or not you could have a life of your own, if you could live for eternity. If you are going to live for eternity, it would seem that you would need to keep finding new things to do, or new ways to be satisfied doing the same things again and again.  Williams’ argument is that you can only talk about such a life as your own life if you remain reasonably close to how you started out. In other words, can it still be your life if it goes on for eternity? Williams’ answer was ‘no’. 

For some philosophers, it is straightforwardly obvious that annihilation, followed by nothingness, is a great and uncompensated evil. Moore quotes the American philosopher Thomas Nagel, who writes that being given the alternatives of living for another week or dying in five minutes would always (all things being equal) opt for living another week. If there were no other catastrophe which could be averted by his death. Which Nagel interprets as being tantamount to wanting to live for ever. He wrote that ‘there is little to be said for death: it is a great curse. If we truly face it, nothing can make it palatable.’ Moore suggests that the opposing points of view of Williams and Nagel may be the consequence of a temperamental difference, as much as an intellectual one.  Nagel also suggested the possibility that Williams might have been more easily bored than he is. Moore says this might have been the case. 

Moore feels that choosing the option of living for another week is not tantamount to wanting to live forever, and that he might choose the option while being appalled at the prospect of living forever. Where does this leave us? Some philosophers celebrate our finite nature, and others lament it. But there is consensus on one point, which is that finitude does not deprive our lives of meaning. Those who celebrated finitude say that it helps to confer meaning. Those who lament it, do so because, in the fullness of time, it threatens to take that meaning away. Moore suggests that what this perhaps shows, is that our finitude is what enables us to reach out and touch the infinite.

John Cottingham argues that there is something about us, though we are obviously animals, obviously biological creatures, which is restless and inclined to reach out. We have transcendent aspirations. What St Augustine called ‘the restlessness of the human heart’. Our finitude fuels our aspirations to something more perfect than we are. 

In the final programme, Moore returns to the words of the mathematician David Hilbert, from a lecture in 1925, on the nature of the infinite, which opened the series:
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 says that what we have seen in the course of this series amply bears out Hilbert’s claim. Philosophers have pondered the vastness of space and time, the limitless of numbers, and the perfection of God. Many however, aware that the idea of the infinite is associated with paradox and mystery, have been disturbed by it, and have been wary of it.  At the root of this division of opinion about the infinite, is our knowledge that we ourselves are finite. 

There is something more fundamental to this than the fact that we are small and ephemeral. We find ourselves limited and constrained by what is utterly independent of us. This gives us a contrasting sense of what is unlimited, unconstrained, and self-sufficient. If scientific investigation should show at some point that space and time are themselves finite, still we would have this sense of the infinite. But it is a sense of something which eludes us. We might say that it is a sense of something which we cannot have any sense of. We are unsure of what to make of the infinite, and consequently unsure of what to make of ourselves.

Like the Greeks two and a half thousand years ago, we find ourselves on the one hand troubled by the infinite, and on the other hand, unable to ignore it. Rene Descartes took issue with the fact that we begin with the finite, and just think of the opposite of it; on the contrary he said that we begin with the infinite, and think of the finite as the opposite of that. And this led him to argue that there must be something in reality, something infinite, from which we get the very idea. Descartes had in mind God. There is a non-theological version of this argument which has survived into contemporary philosophy. Thomas Nagel is quoted on the idea of infinity in connection with numbers:

To get the idea of infinity, we must understand that the numbers we use to count things, are merely the first part of a series that never ends. Though our direct acquaintance with specific numbers is extremely limited, we cannot make sense of it, except by putting them and ourselves, in the context of something larger. Something whose existence is independent of our fragmentary experience of it.  When we think about the finite activity of counting, we come to realise it can only be understood as part of something infinite.

Thomas Nagel finds it striking that we have evolved in a complex way, in such a way that we have a grasp of very powerful objective truths – of logic and mathematics for example, and also certain objective truths of morality, and, according to Nagel, this means that our minds are what he calls ‘instruments of transcendence’. Though we are finite biological creatures, our minds nevertheless reach out to grasp these objective truths, and that is a remarkable fact about us which Nagel thinks can’t be fully explained by the biological processes of mutation and natural selection.

So an atheist such as Nagel nevertheless thinks we are in touch with something transcendent. The topic of this series is the infinite – what does the infinite have to do with what is transcendent? Perhaps these are two quite different ideas. Perhaps the infinite isn’t anything grand at all. It might well be argued, as we have seen in previous programmes, it has been argued, that any time you move from A to B, you do infinitely many things.
Ludwig Wittgenstein tried to take some of the mystique out of the infinite. We naturally think that the infinite, if it exists at all, must be something awesome, and utterly beyond our comprehension. His view was that to understand the infinite, we need to understand phrases, like, ‘and so on’. Simple phrases, which use simple grammatical rules, which finite creatures like ourselves, can easily grasp. “The expression ‘and so on’ is nothing but the expression ‘and so on’. Nothing, that is, but a sign which can’t have meaning but by the rules which have hold of it.”

Moore suggests that, if we accept Wittgenstein’s debunking, it allows us to separate completely the idea of the infinite from the idea of the transcendent.   It allows us to treat the infinite as something quite tame, and unremarkable. Moore should realise this only applies to the potential infinite, and if the idea of an actual infinite, which has connections with the finite world, is baseless.

He continues by suggesting that, whether or not we side with Wittgenstein and admit that there is nothing to the infinite, beyond our finite linguistic resources and the rules governing their use, we must acknowledge our urge to think there is more to the infinite than that. We must concede that we think that there is an infinite with a capital 'I'.  The unconditioned, self-sufficient, and transcendent.

Where does this urge come from?  How does it manifest itself? In conversation with Moore, John Cottingham suggests that there are three categories in our awareness of the beauties of the natural world which seem to have something in them which isn’t just a mere ability to produce pleasure in us – a special quality which is sometimes called a sacred, or even a numinous aspect.  Another area is the requirements of morality, which I recognise as incumbent on me to observe, even though I may not want to.    A third area is what used to be called ‘the eternal truths’ of logic and mathematics, which I recognise as necessary, universal and unalterable. So in all these areas I seem to recognise something which transcends, which goes beyond the ordinary contingent flux of biological and physical circumstances. In so far as I can recognise those areas, I seem to be reaching forward to what might be called 'the dimension of the transcendent'.

Moore comments that we have a sense of the transcendent then, and a sense of the infinite, with the capital ‘I’. This is something akin to a religious experience. Or an experience of the sacred. It suggests something infinitely greater than us, which can give meaning to our lives. But how might such experiences be located in a secular world view? John Cottingham says that these might be understood in terms of weird, funny states we can get into, via psychedelic drugs, or by fasting, which would be a reductionist or deflationary interpretation of the sacred. If we don’t want to go down that route, then we have to say that there is something about reality which calls forth such experiences, and they are not just a private ‘trip’, but a response to something which is real.

The weakest parts of this series tend to be when religion and ideas of what is transcendent are being discussed. The secular and finitist approach is a ball and chain when it comes to talking about the actual infinite.

Moore begins the windup to the series, saying this brings us right back to Descartes, back to the idea that the best explanation for these responses, maybe that they are genuine responses to something which is genuinely infinite, with a capital ‘I’.  

There is however another approach which Moore suggests we can take however, and he gives a striking example of this alternative approach, expressed by the British novelist and philosopher Iris Murdoch. Whereas Descartes had said that our idea of God as something perfect and great, could only have been implanted in us by something real which is in fact so perfect and great.  Iris Murdoch however turned this around, and said our idea of god was so great that nothing in reality could match up. God does not exist, and cannot exist, because something so great could not be confined by mere existence. But what implants the idea in our minds, does exist. And is constantly experienced and pictured. Incarnate in work, knowledge, and love. 

Moore’s judgement of the worth of Murdoch’s formulation is sound. It might have been better to have started the series with this view of the infinite, which does not prioritise existence as a property which is necessary to what is real.  Instead, it makes more sense to understand existence as something which is generated by the Infinite, rather than trying to locate the Infinite within what exists. Which is the principal difference between ancient and modern philosophy, when considering the infinite, and what is real. We look at important questions upside down, and are tempted to re-interpret what came before, in terms of how we think now. Much of our inherited history of ideas is bent out of shape, and sometimes unintelligible to us, having been deformed by many attempts to understand difficult ideas over many centuries.

This particular history of the infinite represents a more or less consensus view of an important aspect of the history of ideas. A consensus in philosophy however, usually means that a serious line of argument has run out of steam, and can tell us nothing new. I think that this is the case here. Moore's account is full of fascinating detail, but the overall narrative is open to question in a number of respects, and it is clear that the story of man's engagement with the infinite could be written in a quite different way.

Thomas Yaeger, October 10, 2017