A Timeline of Texts

My intellectual history is quite complicated. I wrote half a million words between 1986 and 1989, very little of which is currently available (but all of it indexed). Meditations on the Egyptian Ka and Coelum Terrae were both written on an IBM golfball typewriter in 1988, but most other stuff was written in longhand.  A study of Henri Frankfort's view of Parmenides is however available in longhand on this site. 

I was always focussed on ancient history, but I spent a long time studying the English and Italian renaissances as well, which taught me many things which were useful in the study of antiquity. Particularly how to read images.This work was done before I went off to study at UCL in September 1989-92.

I was writing before the advent of a useful internet, and Tim Berners Lee’s World Wide Web. I was aware that what I was writing was not fitted to the contemporary market (which is what most writers and scholars write for these days). This is part of the explanation for the sequence of the following list.

 In 1993, who was going to publish J.G Frazer and the Platonic Theory of Being? Frazer had been heavily criticized since his death, but no-one was looking at his work from my point of view. Most scholars would have agreed with Frazer’s view that a reductionist interpretation of Plato was the right one, even if they disagreed with his methods, and sometimes his interpretations. So that book was finished (over many evenings) in a third floor office in Devonshire Street, London, in 1993, and then parked, because there was nothing I could do with it. For a while at least.

I first gained access to the Internet and the Web in March 1994, at the computing services building in Oxford..I'd learned about what was now possible through a number of Internet magazines on sale in the Science Fiction bookshop in Oxford Street. After that I was always looking for Internet related jobs. I ended up working in Oxford on a Leverhulme funded project focussed on making an author's work available via the web. I wrote a report for the funders and the stakeholders, which mentioned in passing the copyright issues which would arise. That pretty much killed the project as it was originally conceived. But I’d learned a great deal along the way, including how to get Mac files into Word files.

I then moved to a newly minted Encyclopedia company in plush modern offices in South West London, where I built their first proper website. It was an odd sort of company, which attracted the attention of the UK government by not having any visible income at the time. It did eventually get a return on the project, but was initially bankrolled by a member of the government in the United Arab Emirates. I spent a couple of weeks in the UAE along the way.  

In 1997 I moved to the University of Bath as an internet magazine editor and scribe, as well as a manager of web pages and builder of websites. All the staff had personal webspace available to them. I created a website focussed on a well-known poet, and built it up over a few years. It was in existence from 1998 to 2005. Lots of traffic, and questions from students. The files still exist. 

Then in late 1998 I decided to make the book on Frazer available on the Web, partly in response to a request from a Catalan friend, who was (and is) an anthropologist. I didn’t get much feedback from him, which was disappointing, but the access statistics (which I still have) were pretty good. My future course was set.

After that I spent eight years or so as a project manager in Edinburgh, mainly working on institutional repository development, including business analysis and system design. I departed the university in July 2013 in order to write full time.  

Here's the production sequence:

  

Henri Frankfort (et al) on Parmenides. 1987 Unpublished

Meditations on the Egyptian Ka 1987. Unpublished

Coelum Terrae 1988 Unpublished.

Mirrors of the Divine 1991-2 Unpublished

J. G. Frazer and the Platonic Theory of Being 1993. Unpublished on completion. This version was available on the web in html format between New Year’s Day 1999 till September 2005

Magic or Magia? Plato’s Sophist 1994 Unpublished on completion. Available on the Web from 2018 in jpeg format.

The Sacred History of Being 2003-5 initial draft. Unpublished.

Ontology and Representation in Assyria and The Ancient Near East 2005-9. Unpublished.

The Sacred History of Being 2011-14 Published as an ebook in November 2015

J. G. Frazer and the Platonic Theory of Being. Published as an ebook in 2016

Understanding Ancient Thought Published as an ebook in 2017

Man and The Divine Published as an ebook in 2018

Echoes of Eternity Published as an ebook in 2020 (contains a revised version of Mirrors of the Divine plus many other essays written between 2004 and 2020)

Meditations on the Egyptian Ka. 1987 Available on the web in jpeg format in 2020 

Henri Frankfort (et al) on Parmenides. 1987  Available on the web as jpeg files, 2017

Ontology and Representation in Assyria and The Ancient Near East 2005-9. First section published 2021

The Death of Pan  Publication as an ebook in 2021 

The Keys of the Kingdom..Contains new material and revised chapters from The Sacred History of Being 2015, and from later books. Publication as an ebook in 2021.

Items marked in orange are publicly available in some form.Try searching on this site. A number of chapters and articles are also available from institutional repositories, often full-text.

The Tedium of Immortality

Episode nine in the BBC R4 series (2016) 'A History of the Infinite' by the philosopher Adrian Moore) begins with a scene from the opera ‘The Makropulos 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 Makropulos, 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 Makropulous’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. 

Calculus and the Infinitesimals: 'The Ghosts of Departed Quantities'

 

In this  episode of Adrian Moore's 'A History of Infinity' (BBC R4, 2016), 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.

A Sense of Divinity - Descartes and Kant

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Plato and the Transcendental Infinite

 

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

***

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Adrian Moore on Georg Cantor and the Size of Infinity

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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