The Journey To The East by Hermann Hesse (1932)

A slender novella, 88 pages in the Picador paperback version, The Journey To The East is a first-person narrative told by a former member of the secretive ‘League’ of poets, writers and seekers who, in their different ways, all undertook journeys to the East in ‘the troubled, confused, yet so fruitful period following the Great War’ (p.5).

What sets it apart, at least to begin with, is that it is nothing like a sensible factual account of a straightforward ‘journey’ such as you might read by traditional travel writers like Robert Byron or Peter Fleming.

Instead it is more like a fairy story, in which the ‘travellers’ encounter legendary figures and mythical beasts, pass through fictional lands from fables and fairy tales, and travel not only in space, but in time – back into the past, penetrating ‘into the heroic and the magical’ (p.7).

One day, when I was still quite a new member, someone suddenly mentioned that the giant Agramant was a guest in our leaders’ tent, and was trying to persuade them to make their way across Africa in order to liberate some League members from Moorish captivity. Another time we saw the Goblin, the pitch-maker, the comforter, and we presumed that we should make our way towards the Blue Pot.

The giant Agramant, the Goblin. It is fairy land.

Despite these imaginative frills, though, the League feels like a Christian monastic order – casual phrases continually remind the reader that Hesse had an intensely pious Christian upbringing, against which he rebelled but whose stern moral seriousness he kept for the rest of his life.

Thus newcomers to the League are ‘novitiates’, must take an ‘oath’ to renounce the world and its temptations, must wear a ring proclaiming their membership of the order. The journey is referred to as a ‘pilgrimage’ and the travellers as ‘pilgrims’. The leader of the narrator’s group talks freely about ‘grace’ and ‘repentance’, both utterly Christian concepts.

But at the same time it is a phantasmagoria of all the cultural greats through the ages:

Our League was in no way an off-shoot of the post-war years, but that it had extended throughout the whole of world history, sometimes, to be sure, under the surface, but in an unbroken line, that even certain phases of the World War were nothing else but stages in the history of our League; further, that Zoroaster, Lao Tse, Plato, Xenophon, Pythagoras, Albertus Magnus, Don Quixote, Tristram Shandy, Novalis and Baudelaire were co-founders and brothers of our League.

This is a kind of greatest hits of world culture. And the way the ‘pilgrims’ travel is both a physical path or itinerary, very much in the style of medieval pilgrims –

And as we moved on, so had once pilgrims, emperors and crusaders moved on to liberate the Saviour’s grave, or to study Arabian magic; Spanish knights had traveled this way, as well as German scholars, Irish monks and French poets.

But also an imaginative one, as they travel through realms of magic and myth, experiencing not only all times, but the real and the imaginary on the same terms.

The core of the experience, the thing which, looking back, the narrator realises brought him the greatest happiness, was:

The freedom to experience everything imaginable simultaneously, to exchange outward and inward easily, to move Time and Space about like scenes in a theatre.

When you reflect on this, it sounds increasingly like the adventures of someone in their library – with the leisure time to roam freely over time and space, and between factual and imaginative literature.

The plot

The first-person narrator is ‘a violinist and story-teller’ who joined the League with the aim of travelling to the East to meet the princess Fatima and, if possible, to win her love (we learn that all League members have quirky or idiosyncratic goals, one wants to see the coffin of Mohammed, another to learn the Tao).

But the oddest thing about the story is that… they don’t travel to the East. About a third of the way through the text, the narrator tells us that at an early point of the journey, while they were still in Europe, at a place called Morbio Inferiore, a municipality in Switzerland, one of his team’s most loyal servants, Leo, goes missing, so the entire squad sets out to find him, searching up hill and dale.

Not only do they never find him, but his group begins to squabble amongst itself, loses focus. Somehow the journey was abandoned and he never made it to the East. Now, we learn, the narrator is struggling to set it all down in a written account, in a bid to revive the heady joy of those young days.

Now the narrative cuts to ‘the present’, some ten years after the journey. The narrator tells us it is a long time since he was active in the League, he doesn’t know whether it exists any more, he’s not sure it ever existed and these things ever happened to him.

And now the narrator tells us that the episode of missing Leo has given him writer’s block, he doesn’t know how to tell the episode correctly, and can’t manage to get the story past it.

And in an abrupt and surprising switch, the narrative stops being about any journey to the East whatsoever.

Now, surprisingly, the scene cuts back to the narrator’s home town and becomes spectacularly more realistic and mundane. To address his problem of writer’s block, the narrator goes to meet a friend of his who’s a newspaper editor, named Lukas, and who wrote a successful book of war memoirs.

Discussion of the war memoirs gives rise to a consideration of how difficult it is to describe any human experience, at how you need to create eras or characters or plots to even begin to get it down.

Even further than this, how some experiences are so intense or evanescent, that you can’t even be sure you had them. In which case, how do you describe them? Lukas replies that he wrote his book about the war because he simply had to, whether it was any good or not was secondary, the writing itself was vital therapy, which helped him control ‘the nothingness, chaos and suicide’ which would otherwise have overwhelmed him (p.46)

So. This is less a book about a journey anywhere, and a lot more a book about the difficulty of writing a book. Ah.

When the narrator tells Lukas how, in writing his account of the journey to the East, he’s got blocked on this episode of the missing servant, Leo, Lukas promptly looks Leo up in the telephone directory and finds there is a Andreas Leo living at 69a Seilergraben. Maybe it’s the same guy, he says – as if we’re in a 1930s detective novel and not the imaginative phantasmagoria we started out in. ‘Go and see him,’ the editor suggests.

So the narrator does, and finds 69a Seilergraben to be an apartment in an anonymous building in a quiet street. The narrator knocks on the door, questions the neighbours, hangs around, and goes back on successive days. Finally he sees this Leo exit his apartment block and walk quietly to the park where he sits on a bench and eats dried fruit from a tin.

This is not at all the mystical imaginative phantasmagoria I was promised on the back of the book, is it? This is staggeringly mundane.

The narrator approaches Leo, and tries to remind him of their time back in the League and on the great journey East which, the text confirms, happened some 10 years earlier. But Leo is calmly dismissive and walks off, leaving the narrator standing alone in the park as dusk falls, in the rain.

Now he is rejected like this, we learn the narrator is prone to depression, in fact to despair and thoughts of suicide.

I had experienced similar hours in the past. During such periods of despair it seemed to me as if I, a lost pilgrim, had reached the extreme edge of the world, and there was nothing left for me to do but to satisfy my last desire: to let myself fall from the edge of the world into the void — to death. In the course of time this despair returned many times; the compelling suicidal impulse…

In other words, he shows the same bouncing from one to extreme to the other that characterised the Steppenwolf and his moods of suicidal despair. And very like the author himself, a glance at whose biography reveals attempts at suicide, prolonged psychotherapy, and a spell in a mental sanatorium.

The narrator gets home and sits down, still damp from the rain and writes a long letter to Leo, then falls asleep. When he wakes up Leo, is sitting in his living room. Leo reveals he is still a member of the League and says he will take the narrator to see the current President. Leo leads him through the streets of the quiet town by a circuitous route, stopping at various inconsequential locations including a church, to an anonymous building, which is large and labyrinthine on the inside (reminding me of the labyrinthine buildings Franz Kafka’s protagonists stumble through).

The narrator is led into an enormous room full of shelves lined with books which turn out to be the archive the League. Leo suddenly starts singing and, as in movie special effects, the archive recedes into the distance and in the foreground appears a large judgement chamber.

A jury assembles and a ‘Speaker’, who acts like a judge. It has turned into a sort of court-room, which makes the comparison with Kafka feel overwhelming – a confused little man dragged to judgement before a huge, imposing court which he doesn’t understand. The essence of the Kafkaesque.

For the first time the narrator is named as ‘H.H.’. H.H.? So a barely veiled reference to the author himself which, yet again, could barely be more like the Kafka who named his two most famous protagonists K. and Joseph K. with his own initial.

The ‘Speaker’ refers to H.H. as ‘the self-accused’ and asks him:

‘Is your name H.H.? Did you join in the march through Upper Swabia, and in the festival at Bremgarten? Did you desert your colours shortly after Morbio Inferiore? Did you confess that you wanted to write a story of the Journey to the East? Did you consider yourself hampered by your vow of silence about the League’s secrets?’
I answered question after question with ‘Yes’…

So I was expecting H.H. to get hammered, but, surprisingly, he is now given permission to go right ahead and write a full account of the League and all its laws.

He is handed a copy of the manuscript of the Journey he had been working on and which had got bogged down at that moment when Leo left the group. But now, when he rereads it, he feels it is bodged, clumsy, inaccurate and – further – as he tries to amend it, he watches the letters change shape, become patterns and pictures, illegible, the entire manuscript changes form in front of his eyes.

Rather improbably, the Speaker gives him free run of the immense archive to research his book, which leads to a passage where H.H. rummages through the archives to find records about his friends and then himself, but finds the records written in strange languages and arcane scripts. Slowly he realises there isn’t enough time in the world to go through this immense and probably infinite library.

From all sides the unending spaciousness of the archive chamber confronted me eerily. A new thought, a new pain shot threw me like a flash of lightning. I, in my simplicity, wanted to write the story of the League, I, who could not decipher or understand one-thousandth part of those millions of scripts, books, pictures and references in the archives! Humbled, unspeakably foolish, unspeakably ridiculous, not understanding myself, feeling extremely small, I saw myself standing in the midst of this thing with which I had been allowed to play a little in order to make me realize what the League was and what I was myself.

the court magically re-assembles, with the Speaker presiding. Now we learn that this little episode was a further step in H.H.’s trial, to show him how vain and presumptuous his aim of writing a history of the league was. The Speaker asks if he is ready for the verdict on him, and whether he wants it delivered by the Speaker or the President himself.

In a surreal development, the grand figure who emerges from the bloom of the archive hall turns out to be none other than… Leo! The Leo he had followed into the party, who is himself the Leo who was his group’s servant on the Journey and now he comes to think about it, was the same President who initiated him into the League and gave him his ring.

H.H. is covered in shame and confusion. To think that he could write a history of the League. To think that he had imagined the League had ended or had never existed. Now Leo recounts H.H.s sins against the League. Forgetting about its existence. Losing his League ring. Even their long walk through the town had been a test because H.H. should have gone into the church and worshipped, as is fitting, instead of standing outside locked in his impatient egotism. It is his egotism which made him deny the League and sink into a world plagued with depression and despair.

Again, as in so many of Hesse’s books, which you imagine will be about Eastern philosophy, the most eloquent passages are about misery and despair. Leo tells the jury how H.H.s loss of faith in the League led him down into the pit, and delivers some puzzling lines:

‘The defendant did not know until this hour, or could not really believe, that his apostasy and aberration were a test. For a long time he did not give in. He endured it for many years, knowing nothing about the League, remaining alone, and seeing everything in which he believed in ruins. Finally, he could no longer hide and contain himself. His suffering became too great, and you know that as soon as suffering becomes acute enough, one goes forward. Brother H. was led to despair in his test, and despair is the result of each earnest attempt to understand and vindicate human life. Despair is the result of each earnest attempt to go through life with virtue, justice and understanding and to fulfill their requirements. Children live on one side of despair, the awakened on the other side. Defendant H. is no longer a child and is not yet fully awakened. He is still in the midst of despair.’

So: Despair is what you enter when you are no longer a child, when you become a questing adult, and before you are initiated or awakened.

Now President Leo initiates H.H. for a second time, giving him a replacement ring and welcoming him back into the ranks of the League.

This really is nothing at all about any literal Journey To The East, is it? It is about adventures of the spirit, or maybe psychological experiences, in a quiet Swiss town.

Now the President leads H.H. to the final test. He is shown the League archives about himself. Specifically, he is shown several other accounts written by members of his group or party on his Journey of ten years ago. Here he is horrified to read that it is he, H.H. that the other members of the group blamed for Leo’s disappearance, for accusing Leo of having taken key documents with him, it was he, H.H. who was blamed by the rest of the group for spreading dissension.

He learns something about trying to write ‘the truth’ (something which is, to be blunt, fairly obvious), which is that everyone has a different account of what happened, and no ‘truth’ can ever be arrived at.

If the memory of this historian was so very confused and inaccurate, although he apparently made the report in all good faith and with the conviction of its complete veracity – what was the value of my own notes? If ten other accounts by other authors were found about Morbio, Leo and myself, they would presumably all contradict and censure each other.

No, our historical efforts were of no use; there was no point in continuing with them and reading them; one could quietly let them be covered with dust in this section of the archives. ..

How awry, altered and distorted everything and everyone was in these mirrors, how mockingly and unattainably did the face of truth hide itself behind all these reports, counter-reports and legends! What was still truth? What was still credible ?

The final few pages end on an enigmatic moment and symbol. Tucked away in the shelf where his records are stored, he finds a grotesque little statuette, like a pagan idol. Only slowly does he realise it is two-sided, shows two human figures joined at the back. And then slowly makes out that one is a depiction of himself, with blurred features, weak and dying. And as he lights another candle he sees something stirring in the heart of the glass statuette, and realises that some kind of life force is moving from his half of the statuette over into Leo’s

And in the last few sentences of the book he remembers a conversation he had with the servant Leo on the Journey, ten years earlier, amid a wonderful festival early in the journey, where Leo had explained that a pet or writer drains himself in order to give eternal life to his work, just as a mother suckles a baby and gives the babe life, at her own expense. So the poet.

And on this slightly ominous, pregnant image the book ends. The narrator feels very sleepy. He turns to find somewhere to sleep. Maybe enacting exactly the gesture whereby the poet, writer or maker, gives all their spirit and life force to their creation and then expires.

Thoughts

Well, it turns out not to be a literal Journey To The East in the slightest. Anyone expecting a straightforward narrative of a pilgrimage to India will be disappointed and puzzled.

However, anyone familiar with Hesse will be less surprised by its combination of the strangely mundane and the wildly phantasmagorical. This is the same combination as in Steppenwolf, which evolved from being a dull account of a middle-aged boarder in a provincial boarding house into the giddy surrealism of the Magic Theatre.

And Steppenwolf also covered a similar range of emotional or psychological states – to be more precise, it displayed a similar, almost schizophrenic, tendency to jump between extremes of Despair and the giddy heights of ecstatic imaginative delirium.

I had this impression of Hesse as being a lofty propounder of high-minded Eastern philosophy. I wasn’t prepared to encounter so many characters who were so full of despair, self-loathing and so many discussions of suicide.

And I’m still reeling from the way the book is not about a Journey To The East at all; it’s much more about the psychological adventures or journey of a middle-aged man living in a Swiss town. All the key events happen in the narrator’s mind. It is a psychological odyssey.

Building a universe

It’s a small detail, but it’s interesting that Hesse includes among fellow members of the League, not only some of his real-life friends, but characters from his other books.

Thus the character ‘Goldmund’, one of the two leads in Narziss and Goldmund, crops up in his initial memories of the Journey, as does the painter Klingsor, who is the fictional lead of Hesse’s earlier novel Klingsor’s Last Summer.

And when I started reading Hesse’s final novel, The Glass Bead Game, early in the introduction the narrator mentions the League of Journeyers To The East as forerunners of the game. Hesse was quite obviously creating a kind of larger imaginative canon, an imaginarium, in which characters not only from history, not only actual writers and composers, along with mythical and legendary figures, but figures from his own earlier fictions, could meet and mingle on equal terms.


Images of war in The Journey To The East

I am always interested in the social history revealed by older texts. It is striking that Hesse doesn’t just launch straight into his fairy-tale journey, but feels the need to define the times, the era, the period against which his pilgrim is reacting, and that he defines these times by repeated references to the social, economic, cultural and spiritual chaos following Germany’s defeat in the Great War.

Ours have been remarkable times, this period since the World War, troubled and confused, yet, despite this, fertile…

It was shortly after the World War, and the beliefs of the conquered nations were in an extraordinary state of unreality. There was a readiness to believe in things beyond reality…

Have we not just had the experience that a long, horrible, monstrous war has been forgotten, gainsaid, distorted and dismissed by all nations? And now that they have had a short respite, are not the same nations trying to recall by means of exciting war novels what they themselves caused and endured a few years ago?…

At the time that I had the good fortune to join the League – that is, immediately after the end of the World War – our country was full of saviors, prophets, and disciples, of presentiments about the end of the world, or hopes for the dawn of a Third Reich. Shattered by the war, in despair as a result of deprivation and hunger, greatly disillusioned by the seeming futility of all the sacrifices in blood and goods, our people at that time were lured by many phantoms, but there were also many real spiritual advances. There were Bacchanalian dance societies and Anabaptist groups, there was one thing after another that seemed to point to what was wonderful and beyond the veil. There was also at that time a widespread leaning towards Indian, ancient Persian and other Eastern mysteries and religions…

His name is Lukas. He had taken part in the World War and had published a book about it which had a large circulation…

And indeed, from a structural point of view, this editor, Lukas, is included mainly for the discussion he promotes about the struggle he had to write his memoirs of the war, and his eventual conclusion that it was better to write something rather than nothing – even if untrue or less than perfect – if only because the act of writing was so therapeutic and saved him from terrible feelings of despair and suicide.

I’m doing no more than suggest that Hesse, who is generally thought of as a kind of high-minded explorer of timeless values was, in fact, very much a man of his times, and that his thinking was marked and shaped by the great cataclysm which he and his nation lived through just as much as all the other authors of the Weimar period.

Credit

Die Morgenlandfahrt by Hermann Hesse was published in German in 1932. The English translation by Hilda Rosner was published by Peter Owen Ltd in 1956. All references are to the 1995 Picador paperback edition.


Related links

20th century German literature

The Weimar Republic

German history

Alex’s Adventures In Numberland by Alex Bellos (2010)

Alexander Bellos (born in 1969) is a British writer and broadcaster. He is the author of books about Brazil and mathematics, as well as having a column in The Guardian newspaper. After adventures in Brazil (see his Wikipedia page) he returned to England in 2007 and wrote this, his first book. It spent four months in the Sunday Times bestseller list and led on to five more popular maths books.

It’s a hugely enjoyable read for three reasons:

  1. Bellos immediately establishes a candid, open, good bloke persona, sharing stories from his early job as a reporter on the Brighton Argus, telling some colourful anecdotes about his time in Brazil and then being surprisingly open about the way that, when he moved back to Britain, he had no idea what to do. The tone of the book is immediately modern, accessible and friendly.
  2. However this doesn’t mean he is verbose. The opposite. The book is packed with fascinating information. Every single paragraph, almost every sentence contains a fact or insight which makes you sit up and marvel. It is stufffed with good things.
  3. Lastly, although its central theme is mathematics, it approaches this through a wealth of information from the humanities. There is as much history and psychology and anthropology and cultural studies and philosophy as there is actual maths, and these are all subjects which the average humanities graduate can immediately relate to and assimilate.

Chapter Zero – A Head for Numbers

Alex meets Pierre Pica, a linguist who’s studied the Munduruku people of the Amazon and discovered they have little or no sense of numbers. They only have names for numbers up to five. Also, they cluster numbers together logarithmically i.e. the higher the number, the closer together they clustered them. Same thing is done by kindergarten children who only slowly learn that numbers are evenly spaced, in a linear way.

This may be because small children and the Munduruku don’t count so much as estimate using the ratios between numbers.

It may also be because above a certain number (five) Stone Age man needed to make quick estimates along the lines of, Are there more wild animals / members of the other gang, than us?

Another possibility is that distance appears to us to be logarithmic due to perspective: the first fifty yards we see in close detail, the next fifty yards not so detailed, beyond 100 yards looking smaller, and so on.

It appears that we have to be actively taught when young to overcome our logarithmic instincts, and to apply the rule that each successive whole number is an equal distance from its predecessor and successor i.e. the rational numbers lies along a straight line at regular intervals.

More proof that the logarithmic approach is the deep, hard-wired one is the way most of us revert to its perspective when considering big numbers. As John Allen Paulos laments, people make no end of fuss about discrepancies between 2 or 3 or 4 – but are often merrily oblivious to the difference between a million or a billion, let alone a trillion. For most of us these numbers are just ‘big’.

He goes on to describe experiments done on chimpanzees, monkeys and lions which appear to show that animals have the ability to estimate numbers. And then onto experiments with small babies which appear to show that as soon as they can focus on the outside world, babies can detect changes in number of objects.

And it appears that we also have a further number skill, that guesstimating things – the journey takes 30 or 40 minutes, there were twenty or thirty people at the party, you get a hundred, maybe hundred and fifty peas in a sack. When it comes to these figures almost all of us give rough estimates.

To summarise:

  • we are sensitive to small numbers, acutely so of 1, 2, 3, 4, less so of 5, 6, 7, 8, 9
  • left to our own devices we think logarithmically about larger numbers i.e lose the sense of distinction between them, clump them together
  • we have a good ability to guesstimate medium size numbers – 30, 40, 100

But it was only with the invention of notation, a way of writing numbers down, that we were able to create the linear system of counting (where every number is 1 larger than its predecessor, laid out in a straight line, at regular intervals).

And that this cultural invention enabled human beings to transcend our vague guesstimating abilities, and laid the basis for the systematic manipulation of the world which followed

Chapter One – The Counter Culture

The probable origins of counting lie in stock taking in the early agricultural revolution some 8,000 years ago.

We nowadays count using a number base 10 i.e. the decimal system. But other bases have their virtues, especially base 12. It has more factors i.e. is easier to divide: 12 can be divided neatly by 2, 3, 4 and 6. A quarter of 10 is 2.5 but of 12 is 3. A third of 10 is 3.333 but of 12 is 4. Striking that a version of the duodecimal system (pounds, shillings and pence) hung on in Britain till we finally went metric in the 1970s. There is even a Duodecimal Society of America which still actively campaigns for the superiority of a base 12 counting scheme.

Bellos describes a bewildering variety of other counting systems and bases. In 1716 King Charles XII of Sweden asked Emmanuel Swedenborg to devise a new counting system with a base of 64. The Arara in the Amazon count in pairs, the Renaissance author Luca Paccioli was just one of hundreds who have devised finger-based systems of counting – indeed, the widespread use of base 10 probably stems from the fact that we have ten fingers and toes.

He describes a complicated Chinese system where every part of the hand and fingers has a value which allows you to count up to nearly a billion – on one hand!

The Yupno system which attributes a different value for parts of the body up to its highest number, 33, represented by the penis.

Diagram showing numbers attributed to parts of the body by the Yupno tribe

Diagram showing numbers attributed to parts of the body by the Yupno tribe

There’s another point to make about his whole approach which comes out if we compare him with the popular maths books by John Allen Paulos which I’ve just read.

Paulos clearly sees the need to leaven his explanations of comparative probability and Arrow’s Theorem and so on with lighter material and so his strategy is to chuck into his text things which interest him: corny jokes, anecdotes about baseball, casual random digressions which occur to him in mid-flow. But al his examples clearly 1. emanate from Paulos’s own interests and hobby horses (especially baseball) and 2. they are tacked onto the subjects being discussed.

Bellos, also, has grasped that the general reader needs to be spoonfed maths via generous helpings of other, more easily digestible material. But Bellos’s choice of material arises naturally from the topic under discussion. The humour emerges naturally and easily from the subject matter instead of being tacked on in the form of bad jokes.

You feel yourself in the hands of a master storyteller who has all sorts of wonderful things to explain to you.

In fourth millennium BC, an early counting system was created by pressing a reed into soft clay. By 2700 BC the Sumerians were using cuneiform. And they had number symbols for 1, 10, 60 and 3,600 – a mix of decimal and sexagesimal systems.

Why the Sumerians grouped their numbers in 60s has been described as one of the greatest unresolved mysteries in the history of arithmetic. (p.58)

Measuring in 60s was inherited by the Babylonians, the Egyptians and the Greeks and is why we still measure hours in 60 minutes and the divisions of a circle by 360 degrees.

I didn’t know that after the French Revolution, when the National Convention introduced the decimal system of weights and measures, it also tried to decimalise time, introducing a new system whereby every day would be divided into ten hours, each of a hundred minutes, each divided into 100 seconds. Thus there were a very neat 10 x 100 x 100 = 100,000 seconds in a day. But it failed. An hour of 60 minutes turns out to be a deeply useful division of time, intuitively measurable, and a reasonable amount of time to spend on tasks. The reform was quietly dropped after six months, although revolutionary decimal clocks still exist.

Studies consistently show that Chinese children find it easier to count than European children. This may be because of our system of notation, or the structure of number names. Instead of eleven or twelve, Chinese, Japanese and Koreans say the equivalent of ten one, ten two. 21 and 22 become two ten one and two ten two. It has been shown that this makes it a lot simpler and more intuitive to do basic addition and subtraction.

Bellos goes on to describe the various systems of abacuses which have developed in different cultures, before explaining the phenomenal popularity of abacus counting, abacus clubs, and abacus championships in Japan which helps kids develop the ability to perform anzan, using the mental image of an abacus to help its practitioners to sums at phenomenal speed.

Chapter Two – Behold!

The mystical sense of the deep meaning of numbers, from Pythagoras with his vegetarian religious cult of numbers in 4th century BC Athens to Jerome Carter who advises leading rap stars about the numerological significance of their names.

Euclid and the elegant and pure way he deduced mathematical theorems from a handful of basic axioms.

A description of the basic Platonic shapes leads into the nature of tessalating tiles, and the Arab pioneering of abstract design. The complex designs of the Sierpinski carpet and the Menger sponge. And then the complex and sophisticated world of origami, which has its traditionalists, its pioneers and surprising applications to various fields of advanced science, introducing us to the American guru of modern origami, Robert Lang, and the Japanese rebel, Kazuo Haga, father of Haga’s Theorem.

Chapter Three – Something About Nothing

A bombardment of information about the counting systems of ancient Hindus, Buddhists, about number symbols in Sanskrit, Hebrew, Greek and Latin. How the concept of zero was slowly evolved in India and moved to the Muslim world with the result that the symbols we use nowadays are known as the Arabic numerals.

A digression into ‘a set of arithmetical tricks known as Vedic Mathematics ‘ devised by a young Indian swami at the start of the twentieth century, Bharati Krishna Tirthaji, based on a series of 16 aphorisms which he found in the ancient holy texts known as the Vedas.

Shankaracharya is a commonly used title of heads of monasteries called mathas in the Advaita Vedanta tradition. Tirthaji was the Shankaracharya of the monastery at Puri. Bellos goes to visit the current Shankaracharya who explains the closeness, in fact the identity, of mathematics and Hindu spirituality.

Chapter Four – Life of Pi

An entire chapter about pi which turns out not only to be a fundamental aspect of calculating radiuses and diameters and volumes of circles and cubes, but also to have a long history of mathematicians vying with each other to work out its value to as many decimal places as possible (we currently know the value of pi to 2.7 trillion decimal places) and the surprising history of people who have set records reciting the value if pi.

Thus, in 2006, retired Japanese engineer Akira Haraguchi set a world record for reciting the value of pi to the first 100,000 decimal places from memory! It took 16 hours with five minute beaks every two hours to eat rice balls and drink some water.

There are several types or classes of numbers:

  • natural numbers – 1, 2, 3, 4, 5, 6, 7…
  • integers – all the natural numbers, but including the negative ones as well – …-3, -2, -1, 0, 1, 2, 3…
  • fractions
  • which are also called rational numbers
  • numbers which cannot be written as fractions are called irrational numbers
  • transcendent numbers – ‘a transcendental number is an irrational number that cannot be described by an equation with a finite number of terms’

The qualities of the heptagonal 50p coin and the related qualities of the Reuleux triangle.

Chapter Five – The x-factor

The origin of algebra (in Arab mathematicians).

Bellos makes the big historical point that for the Greeks (Pythagoras, Plato, Euclid) maths was geometric. They thought of maths as being about shapes – circles, triangles, squares and so on. These shapes had hidden properties which maths revealed, thus giving – the Pythagoreans thought – insight into the secret deeper values of the world.

It is only with the introduction of algebra in the 17th century (Bellos attributes its widespread adoption to Descartes’s Method in the 1640s) that it is possible to fly free of shapes into whole new worlds of abstract numbers and formulae.

Logarithms turn the difficult operation of multiplication into the simpler operation of addition. If X x Y = Z, then log X + log Y = log Z. They were invented by a Scottish laird John Napier, and publicised in a huge book of logarithmic tables published in 1614. Englishman Henry Briggs established logarithms to base 10 in 1628. In 1620 Englishman Edmund Gunter marked logarithms on a ruler. Later in the 1620s Englishman William Oughtred placed two logarithmic rulers next to each other to create the slide rule.

Three hundred years of dominance by the slide rule was brought to a screeching halt by the launch of the first pocket calculator in 1972.

Quadratic equations are equations with an x and an x², e.g. 3x² + 2x – 4 = 0. ‘Quadratics have become so crucial to the understanding of the world, that it is no exaggeration to say that they underpin modern science’ (p.200).

Chapter Six – Playtime

Number games. The origin of Sudoku, which is Japanese for ‘the number must appear only once’. There are some 5 billion ways for numbers to be arranged in a table of nine cells so that the sum of any row or column is the same.

There have, apparently, only been four international puzzle crazes with a mathematical slant – the tangram, the Fifteen puzzle, Rubik’s cube and Sudoku – and Bellos describes the origin and nature and solutions to all four. More than 300 million cubes have seen sold since Ernö Rubik came up with the idea in 1974. Bellos gives us the latest records set in the hyper-competitive sport of speedcubing: the current record of restoring a copletely scrambled cube to order (i.e. all the faces of one colour) is 7.08 seconds, a record held by Erik Akkersdijk, a 19-year-old Dutch student.

A visit to the annual Gathering for Gardner, honouring Martin Gardner, one of the greatest popularisers of mathematical games and puzzles who Bellos visits. The origin of the ambigram, and the computer game Tetris.

Chapter Seven – Secrets of Succession

The joy of sequences. Prime numbers.

The fundamental theorem of arithmetic – In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers.

The Goldbach conjecture – one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that, Every even integer greater than 2 can be expressed as the sum of two primes. The conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort.

Neil Sloane’s idea of persistence – The number of steps it takes to get to a single digit by multiplying all the digits of the preceding number to obtain a second number, then multiplying all the digits of that number to get a third number, and so on until you get down to a single digit. 88 has a persistence of three.

88 → 8 x 8 = 64 → 6 x 4 = 24 → 2 x 4 = 8

John Horton Conway’s idea of the powertrain – For any number abcd its powertrain goes to abcd, in the case of numbers with an odd number of digits the final one has no power, abcde’s powertrain is abcde.

The Recamán sequence Subtract if you can, unless a) it would result in a negative number or b) the number is already in the sequence. The result is:

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11….

Gijswijt’s sequence a self-describing sequence where each term counts the maximum number of repeated blocks of numbers in the sequence immediately preceding that term.

1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, …

Perfect number A perfect number is any number that is equal to the sum of its factors. Thus 6 – its factors (the numbers which divided into it) are 1, 2 and 3. Which also add up to (are the sum of) 6. The next perfect number is 28 because its factors – 1, 2, 4, 7, 14 – add up to 28. And so on.

Amicable numbers A number is amicable if the sum of the factors of the first number equals the second number, and if the sum of the factors of the second number equals the first. The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Added together these make 284. The factors of 284 are 1, 2, 4, 71 and 142. Added together they make 220!

Sociable numbers In 1918 Paul Poulet invented the term sociable numbers. ‘The members of aliquot cycles of length greater than 2 are often called sociable numbers. The smallest two such cycles have length 5 and 28’

Mersenne’s prime A prime number which can be written in the form 2n – 1 a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, … and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, …

These and every other sequence ever created by humankind are documented on The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane’s. This is an online database of integer sequences, created and maintained by Neil Sloane while a researcher at AT&T Labs.

Chapter Eight – Gold Finger

The golden section a number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.

Phi The number is often symbolized using phi, after the 21st letter of the Greek alphabet. In an equation form:

a/b = (a+b)/a = 1.6180339887498948420 …

As with pi (the ratio of the circumference of a circle to its diameter), the digits go on and on, theoretically into infinity. Phi is usually rounded off to 1.618.

The Fibonnaci sequence Each number in the sequence is the sum of the two numbers that precede it. So the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it is Xn+2= Xn+1 + Xn.

as the basis of seeds in flowerheads, arrangement of leaves round a stem, design of nautilus shell and much more.

Chapter Nine – Chance Is A Fine Thing

A chapter about probability and gambling.

Impossibility has a value 0, certainty a value 1, everything else is in between. Probabilities can be expressed as fractions e.g. 1/6 chance of rolling a 6 on a die, or as percentages, 16.6%, or as decimals, 0.16…

The probability is something not happening is 1 minus the probability of that thing happening.

Probability was defined and given mathematical form in 17th century. One contribution was the questions the Chevalier de Méré asked the mathematical prodigy Blaise Pascal. Pascal corresponded with his friend, Pierre de Fermat, and they worked out the bases of probability theory.

Expected value is what you can expect to get out of a bet. Bellos takes us on a tour of the usual suspects – rolling dice, tossing coins, and roulette (invented in France).

Payback percentage if you bet £10 at craps, you can expect – over time – to receive an average of about £9.86 back. In other words craps has a payback percentage of 98.6 percent. European roulette has a payback percentage of 97.3 percent. American roulette, 94.7 percent. On other words, gambling is a fancy way of giving your money away. A miserly slot machine has a payback percentage of 85%. The National Lottery has a payback percentage of 50%.

The law of large numbers The more you play a game of chance, the more likely the results will approach the statistical probability. Toss a coin three times, you might get three heads. Toss a coin a thousand times, the chances are you will get very close the statistical probability of 50% heads.

The law of very large numbers With a large enough sample, outrageous coincidences become likely.

The gambler’s fallacy The mistaken belief that, if something happens more frequently than normal during a given period, it will happen less frequently in the future (or vice versa). In other words, that a random process becomes less random, and more predictable, the more it is repeated.

The birthday paradox The probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. (These conclusions are based on the assumption that each day of the year (excluding February 29) is equally probable for a birthday.) In other words you only need a group of 23 people to have an evens chance that two of them share a birthday.

The drunkard’s walk

The difficulty of attaining true randomness and the human addiction to finding meaning in anything.

The distinction between playing strategy (best strategy to win a game) and betting strategy (best strategy to maximise your winnings), not always the same.

Chapter Ten – Situation Normal

Carl Friedrich Gauss, the bell curve, normal distribution aka Gaussian distribution. Normal or Gaurrian distribution results in a bell curve. Bellos describes the invention and refinement of the bell curve (he explains that ‘the long tail’ results from a mathematician who envisioned a thin bell curve as looking like two kangaroos facing each other with their long tails heading off in opposite directions). And why

Regression to the mean – if the outcome of an event is determined at least in part by random factors, then an extreme event will probably be followed by one that is less extreme. And recent devastating analyses which show how startlingly random sports achievements are, from leading baseball hitters to Simon Kuper and Stefan Szymanski’s analysis of the form of the England soccer team.

Chapter Eleven – The End of the Line

Two breakthroughs which paved the way for modern i.e. 20th century, maths: the invention of non-Euclidean geometry, specifically the concept of hyperbolic geometry. To picture this draw a triangle on a Pringle. it is recognisably a triangle but all its angles do not add up to 180°, therefore it defies, escapes, eludes all the rule of Euclidean geometry, which were designed for flat 2D surfaces.

Bellos introduces us to Daina Taimina, a maths prof at Cornell University, who invented a way of crocheting hyperbolic surfaces. The result looks curly, like curly kale or the surface of coral.

Anyway, the breakaway from flat 2-D Euclidean space led to theories about curved geometry, either convex like a sphere, or hyperbolic like the pringle. It was this notion of curved space, which paved the way for Einstein’s breakthrough ideas in the early 20th century.

The second big breakthrough was Georg Cantor’s discovery that you can have many different types of infinity. Until Cantor the mathematical tradition from the ancient Greeks to Galileo and Newton had fought shy of infinity which threatened to disrupt so many formulae.

Cantor’s breakthrough was to stop thinking about numbers, and instead think of sets. This is demonstrated through the paradoxes of Hilbert’s Hotel. You need to buckle your safety belt to understand it.

Thoughts

This is easily the best book about maths I’ve ever read. It gives you a panoramic history of the subject which starts with innumerate cavemen and takes us to the edge of Einstein’s great discoveries. But Bellos adds to it all kinds of levels and abilities.

He is engaging and candid and funny. He is fantastically authoritative, taking us gently into forests of daunting mathematical theory without placing a foot wrong. He’s a great explainer. He knows a good story when he sees one, and how to tell it engagingly. And in every chapter there is a ‘human angle’ as he describes his own personal meetings and interviews with many of the (living) key players in the world of contemporary maths, games and puzzles.

Like the Ian Stewart book but on a vastly bigger scale, Bellos makes you feel what it is like to be a mathematician, not just interested in nature’s patterns (the basis of Stewart’s book, Nature’s Numbers) but in the beauty of mathematical theories and discoveries for their own sakes. (This comes over very strongly in chapter seven with its description of some of the weirdest and wackiest number sequences dreamed up by the human mind.) I’ve often read scientists describing the beauty of mathematical theories, but Bellos’s book really helps you develop a feel for this kind of beauty.

For me, I think three broad conclusions emerged:

1. Most mathematicians are in it for the fun. Setting yourself, and solving, mathematical puzzles is obviously extremely rewarding. Maths includes the vast territory of puzzles and games, such as the Sudoku and so on he describes in chapter six. Obviously it has all sorts of real-world application in physics, engineering and so on, but Bellos’s book really brings over that a true understanding of maths begins in puzzles, games and patterns, and often remains there for a lifetime. Like everything else maths is no highly professionalised the property of tenured professors in universities; and yet even to this day – as throughout its history – contributions can be made by enthusiastic amateurs.

2. As he points out repeatedly, many insights which started out as the hobby horses of obsessives, or arcane breakthroughs on the borders of our understanding, and which have been airily dismissed by the professionals, often end up being useful, having applications no-one dreamed of. Either they help unravel aspects of the physical universe undreamed of when they were discovered, or have been useful to human artificers. Thus the development of random number sequences seemed utterly pointless in the 19th century, but now underlies much internet security.

On a profounder note, Bellos expresses the eerie, mystical sense many mathematicians have that it seems so strange, so pregnant with meaning, that so many of these arcane numbers end up explaining aspects of the world their inventors knew nothing of. Ian Stewart has an admirably pragmatic explanation for this: he speculates that nature uses everything it can find in order to build efficient life forms. Or, to be less teleological, over the past 3 and a half billion years, every combination of useful patterns has been tried out. Given this length of time, and the incalculable variety of life forms which have evolved on this planet, it would be strange if every number system conceivable by one of those life forms – humankind – had not been tried out at one time or another.

3. My third conclusion is that, despite John Allen Paulos’s and Bellos’s insistence, I do not live in a world ever-more bombarded by maths. I don’t gamble on anything, and I don’t follow sports – the two biggest popular areas where maths is important – and the third is the twin areas of surveys and opinion polls (55% of Americans believe in alien abductions etc etc) and the daily blizzard of reports (for example, I see in today’s paper that the ‘Number of primary school children at referral units soars’).

I register their existence but they don’t impact on me for the simple reason that I don’t believe any of them. In 1992 every opinion poll said John Major would lose the general election, but he won with a thumping majority. Since then I haven’t believed any poll about anything. For example almost all the opinion polls predicted a win for Remain in the Brexit vote. Why does any sane person believe opinion polls?

And ‘new and shocking’ reports come out at the rate of a dozen a day and, on closer examination, lots of them turn out to be recycled information, or much much more mundane releases of data sets from which journalists are paid to draw the most shocking and extreme conclusions. Some may be of fleeting interest but once you really grasp that the people reporting them to you are paid to exaggerate and horrify, you soon learn to ignore them.

If you reject or ignore these areas – sport, gambling and the news (made up of rehashed opinion polls, surveys and reports) – then unless you’re in a profession which actively requires the sophisticated manipulation of figures, I’d speculate that most of the rest of us barely come into contact with numbers from one day to the next.

I think that’s the answer to Paulos and Bellos when they are in their ‘why aren’t more people mathematically numerate?’ mode. It’s because maths is difficult, and counter-intuitive, and hard to understand and follow, it is a lot of work, it does make your head ache. Even trying to solve a simple binomial equation hurt my brain.

But I think the biggest reason that ‘we’ are so innumerate is simply that – beautiful, elegant, satisfying and thought-provoking though maths may be to the professionals – maths is more or less irrelevant to most of our day to day lives, most of the time.


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