A History of the Infinite (Adrian Moore)

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 19th September. Episodes are discussed in order of broadcast.

I had an exchange with Adrian Moore halfway through the series, in which he brought up some issues, and I responded to these the following day. I suggested that the content of this exchange would be reflected in a coda to the series of reviews, once all of the programmes had been written up.

Since then, I have decided to write a full review of the series, since Moore's case depends on pressing into service the ontological argument, which I've shown to be problematic elsewhere, and though it claims to provide a rational basis for belief in God, it does no such thing. Using the medieval ontological argument necessarily diverts us from the ancient modes of discussion concerning the divine and the infinite. The result is, as in this series, a warped view of the history of human thought.

[updated February 19, 2017].

The rest of the commentary was commenced again on November 13. All ten summaries of the programmes are now available below.

TY, November 17, 2016

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(The podcasts are available by clicking the image)

One: Horror of the Infinite

The first programme begins with the words of David Hilbert from a lecture in 1925, on the nature of the infinite. I include here the sentence which precedes the quotation used by Adrian Moore:

"The definitive clarification of the nature of the infinite has become necessary, not merely for the special interests of the individual sciences, but rather for the honour of the human understanding itself. 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."

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.

Our story begins in the 6th century BC, with the Greek mathematician and philosopher Pythagoras…”

So, Moore has set out his stall. ‘Ever since people have been able to reflect’ clearly echoes the standard classicists view that sophisticated abstract thought began with the Greeks, and that the development of abstract thought was a secular matter, and nothing to do with religion or religious ideas. He does acknowledge that the idea of infinity or the unlimited (apeiron) has at times been discussed in connection with the divine, but by saying this he is looking forward to his programme on the medieval ontological argument. Argument about infinity in antiquity has nothing to do with cultic thought and behaviour, which is treated by classicists mostly as an irrational, pathological or delusional set of phenomena. Or otherwise can be understood as reflecting the social dynamics of the society in which cult activity is found. So we needn’t bother to try to look inside: describing the phenomena in context is enough.

He then discusses Pythagoras and the beliefs of his followers, who he acknowledges to comprise a religious group (it is not possible to deny this). But the pairs of opposites that he repeats here, which formed part of Pythagorean doctrine, have nothing to do with the gods or the divine. Instead they are evidence of their belief that the universe consisted of a war between these opposites.

It is still acceptable practice for classicists and historians of philosophy to studiously ignore Mesopotamia and Egypt when discussing the history of thought (not all of them are guilty of this). For some of us, who studied the Ancient Near East at the same time as we studied Greece, what the classicists accept as axiomatically true (that Greece is the home of philosophy, and the development of abstract thought) isn’t true at all. But the classicists aren’t listening. The Shen symbol for example, found in Egypt art, and also in Mesopotamian art, clearly indicates the concept of infinity, and its use stretches far into the past. We are also told by later Greek writers that some key aspects of Pythagorean doctrine, concerning abstraction and infinity, were borrowed from the Babylonians. So it is a form of scholarly self-indulgence to continue to retell the history of infinity as if the idea dates only from the 6th century BCE.

Moore argues that the Pythagoreans thought finite things were good, and that infinite things were bad, 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 – except that Moore refers to rational numbers as ‘counting numbers’. Which shows how dumbed down this series of programmes is going to be. The terms ‘rational numbers’ and ‘irrational numbers’ are not used, even when they are being talked about.

Moore wheels on Richard Sorabji to talk about 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.

The programme moves on to consider whether other ancient Greeks had the same resistance to the infinite. Richard Sorabji makes his second appearance, to talk about the views of Anaxagoras, on infinite divisibility. Anaxagoras was comfortable about these ideas. Moore then talks about Zeno’s paradoxes in connection with infinite divisibility. Richard Sorabji appears again to explain the paradox of travelling by an infinite number of half distances, which seems to imply that movement is impossible. Ursula Coope then outlines the similar paradox of Achilles and the Tortoise.

Observation and reflection thus seem to contradict each other. Zeno distrusted observation. Sorabji appeared once again in connection with  Zeno’s view 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. 

TY September 19, 2016

Two: Aristotle’s Rapprochment

It may be surprising that the second episode of Adrian Moore’s account of the history of Infinity does not deal with Plato, except to explain that Aristotle was a pupil at Plato’s Academy, and to mention the famous dictum above the door, that ‘no-one should be admitted who was ignorant of geometry’. But that is the case. The reason may be the difficulty in understanding what Plato meant about almost anything (one recent translator of Plato’s Philebus admitted in his preface that he did not understand what the dialogue was about).

However this is a telling omission from this history, since 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 only reference to Parmenides and his notion of the universe as simply one and one alone, is as an introduction in episode one to his pupil Zeno of Elea, and his response to paradox. There is no discussion of Plato’s demolition of Parmenides arguments, and 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), which is a way of understanding the relationship of finite things to the infinite.

Instead, by moving on to Aristotle, Moore explores his response to the paradoxes of Zeno, against which Aristotle thought he had important arguments.

Both Richard Sorabji and Ursula Coope reappear in this second episode to give authority to the discussion concerning the views of Aristotle and Zeno.

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. 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. There was some pointless dramatization of a journey home by Moore here, involving two rainstorms, which didn’t add anything to the discussion. 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 example given by Ursula Coope was the limitation of the human form, which would not change if all of the other things in our environment were removed. Our form is not limited or defined by those things). 

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.

It was 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.

That was the end of the argument. There was one glaring error of detail – Aristotle was not a third century BCE figure, but died a year after his pupil Alexander the Great (d. 323). So Aristotle was dead before the third century started. Moore has been rehearsing his history of infinity for a long time, so that shouldn’t have happened.

Of course this history of infinity is written and conceived from a secular standpoint, since the story of it is understood as one of many strands of the history of philosophical ideas. Leaving Plato out of the argument makes it much easier to ignore the shadow of the idea of both finite and infinite things in Greek cultic life. Plato’s forms are clearly finite things (though of an abstract nature), which also have communion with the infinite (The Good).

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. Perhaps a distinction Aristotle did not invent, but possibly borrowed from Platonic teaching in the Academy, since Aristotle was familiar with the theory of the forms.

The next episode jumps to Giordano Bruno. Wow!

TY September 20, 2016

Three: The Infinite and the Divine

After having skipped a discussion of Plato in part two of this series of broadcasts, it might seem surprising that Adrian Moore’s first port of call in part three is the philosopher Plotinus, who was writing in the third century CE, 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. Plato’s principal interest was in a transcendent reality, which it would be hard to distinguish from the infinite. 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.

The Babylonians had a story (preserved for us by the priest Berossus) about the meeting of one of their sages (apkallu) with an amphibious creature (of horrible aspect) who emerged from the sea during the day, and retired back into the waters at night. This creature, Oannes, revealed the knowledge of the crafts, of agriculture, irrigation, and of land measure, and so on. For the Babylonians (and the Greeks for that matter), water, the sea, and the ocean, were all images of the unlimited, and unimaginable and unmeasurable vastness. Water is all around the world (in a circle, as both the Greeks and the Babylonians imagined), is colourless and transparent, and gives life to finite creatures such as man, and also to the plants and animals which he needs for his survival. The import of the story is that it was contemplation of the unlimited that gave rise to knowledge, both of the infinite, and of finite things.

One might also recall the statement of Yahweh in the Old Testament, in which he says that he is ‘the first and the last, and there is no other god before me’. In the book of Malachi, Yahweh says ‘I do not change’. That is an explicitly philosophical description of an infinity which is all-pervasive, and is present all at once, and is a prime characteristic of the divinity of the Hebrews.

It is possible to amplify in numerous ways the antiquity of the connection between the idea of the infinite and the idea of the divine, including by reference to details of the Babylonian creation story, but I think the point is made.

Moore argues that Plotinus’ idea that the divine was infinity itself was taken up by Thomas Aquinas (spool forward several hundred years to the early Middle Ages). Some background to Aquinas was delivered by Cecilia Trifogli. 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.

Not everyone agreed with what became the doctrinal position. Spool forward another few hundred years to the arrest of the scholar Giordano Bruno in the late sixteenth century, who was then 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. The number of numbers in existence was discussed, 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. 

TY, September 21, 2016

Four: The Infinite and Human Experience

The fourth programme skips on again in time in order to discuss the views of Rene Descartes in the sixteenth century, and 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 fourth century BCE (Aristotle, Zeno), the third century CE (Plotinus), and the 13th century CE (Aquinas). Which is a journey of sixteen hundred years.

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 about the role and history of infinity in mathematics and in physics, and the fascinating paradoxes and problems which later investigation has thrown up. The first episodes are therefore a necessary preamble 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. Again, a telling quotation, which hints at the richness and strangeness of the unexplored territory between the fourth century BCE and the thirteenth CE, 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 this 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. 

So Descartes idea of his own finite reality was in fact 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. 

Where did this idea of knowledge by description in connection with an understanding of the nature of the divine come from? Medieval Kabbalists used exactly the same argument to justify their use of a limited set of the properties and attributes of the divine, because as limited beings, it would be impossible to fully understand God. Descartes by his own admission read some strange books early in his philosophical career, and he may well have studied the Kabbalah (other philosophers did).

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. So it was rather odd to hear Berkeley described as an empiricist.

Finally Moore discusses a narrow aspect of Kant’s understanding of the idea of infinity. This final part of the episode represents a mangled and highly misleading understanding of Kant, which I will discuss in a few moments. 

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.

At this point John Cottingham talks about his understanding of Kant, which is that for him knowledge is confined to the five senses, or we leave solid ground and end up in metaphysics. Kant 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.

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). The moral law is what ought to direct us in all we do, with infinite respect granted to fellow rational beings.

This is not the world of the five senses or space and time. So Moore and Cottingham do not seem to agree about what Kant thought was important.

One of Kant’s principle 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 Reason, 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 through the categories of our understanding. So 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. But crucially, he, as an Enlightenment figure, did not read Plato’s understanding of reality in terms of how the infinite and the divine might be approached. In those days no-one did. It is still treated as something to be avoided. 

TY September 22, 2016

Five: The Mathematics of the Infinitely Small

This episode is about the nature and development of the calculus. It begins with the observation (dramatized with a conversation between a child and the online robot Siri) 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 talked about is lacking in the kind of precise description you might expect.

A train travelling at a regular speed is used as an illustration. Travelling a distance of sixty miles over an hour means that the train had a 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 full title of his book is ‘The Analyst, subtitled: A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith.’

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 alludes to 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.”

TY September 23, 2016

Six: The Mathematics of the Infinitely Big

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 this. But that was all he accepted. Aristotle 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 asks the questions – 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.

How do you count without actually counting, and know if a set or collection is the same size as another? Mary Leng (University of York) explained that 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.

Mary Leng explained that 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.

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’

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. 

TY, September 27, 2016

Seven: Crisis and Uncertainty

Georg Cantor created a formal theory of the infinite in the late nineteenth century. Moore says that this theory was a work of genius. Some aspects of this theory were discussed in the last programme. Cantor’s work however, took its toll, and he eventually died in an asylum. 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. Cantor proved that some infinities are bigger than others (at least in mathematical terms. This is a question of representation to the human understanding, and of the categories implicit in how we understand things, since the definition of the actual infinite is quite clear. The implication of bigger infinities, is that some infinities are less than infinite. Infinity itself however, is not to be measured in scalar or spatial terms. TY). He proved that there are always more sets of things, than there are individual things of that kind.

Marcus Giaquinto (UCL) then has a brief slot, where he tells us essentially what Moore just told us, that Cantor proved there are always more sets of things than things. Then back to Moore, who invites us to consider the issue of sets of sets. How can there be more 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. Second outing for Marcus Giaquinto, in conversation with Moore. 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. Third outing for Marcus Giaquinto, who confirms that it looks as though we are stuck with an unanswerable question.  He thinks it is 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 Godel 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.

Conclusion - 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. Fourth outing for Marcus Giaquinto, who points out that 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. Godel’s work however, made it unlikely that this programme would be a success.

Fifth outing for Marcus Giaqunto, who argues that there is not a crisis in modern mathematics, and 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. 

TY November 13, 2016

Eight: The Cosmos

This 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 (4th 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 is however 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? 
Enter Jo Dunkley (Oxford Astrophysics), who explains how 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.

Moore asks  the question: if space were curved like the surface of the earth, how would that manifest itself? Jo Dunkley then distinguishes 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. Jo Dunkley says that 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. John Barrow says that 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. Barrow suggests that the 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, Barrow points out.  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. 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.

TY November 14, 2016

Nine: Death and Immortality

The programme 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 they did not fear or deplore death. They did not see how they should be affected by something they would not be around to witness. Enter Sophie Chappell (Open University) who elaborated on the Epicurean argument, by saying that 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 inoccuousness 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. Second outing for Sophie Chappell, who says that 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’. 

Chappell disagrees with Williams for two reasons – the first is that she is not so sure that it would be impossible to keep finding new things to do, or new ways of doing the same things, and the second reason is that she is not as sure as Williams is that you need to stay as the same person. Chappell uses the images of seeds and plants, and caterpillars and butterflies – the seed turns into a plant, and the caterpillar turns into a butterfly, and suggests it may be the same with human beings.

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. Moore says that this was the view of Rene Descartes.

Enter John Cottingham, who appeared in one of the earlier programmes in this series. 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. 

The next and final programme looks at how our self-consciousness about our own limitations, gives the great questions of the infinite their urgency.

Ten: Where does this leave us?

In this 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.

Enter John Cottingham again, who begins by suggesting that Descartes point of view was a materialist one, quickly qualifying this suggestion by saying that Descartes did not have anything like a traditional belief in God, he nevertheless finds it striking that we have evolved [he didn’t think that!] 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. [I’m wondering here if there has been a poor edit of what John Cottingham said, so that his comment about Descartes is immediately followed by a discussion of Nagel’s view].

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 [described at this point as German, though he was in fact Austrian] 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.
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. 

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. Moore asks the question: where does this urge come from?  How does it manifest itself? The second outing for John Cottingham, who says 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? Another contribution from John Cottingham, who 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.

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 we can take however. For 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. 

TY, November 17, 2016

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