Siddhartha by Hermann Hesse (1922)

Siddhartha is a brief (119-page) telling of the life story of a (fictional) contemporary of the Buddha, a fellow seeker after truth and spiritual enlightenment. The book describes his life and experiences as he follows his own personal path to enlightenment.

Siddhartha is told in simple, lucid prose and has, from start to finish, the feel of a fable, or of a certain kind of old-fashioned children’s story.

I read it in the beautifully clear and rhythmic English translation by Hilda Rosner, which was first published in 1951.

In the shade of the house, in the sunshine of the riverbank near the boats, in the shade of the Salwood forest, in the shade of the fig tree is where Siddhartha grew up, the handsome son of the Brahman, the young falcon, together with his friend Govinda, son of a Brahman. The sun tanned his light shoulders by the banks of the river when bathing, performing the sacred ablutions, the sacred offerings. In the mango grove, shade poured into his black eyes, when playing as a boy, when his mother sang, when the sacred offerings were made, when his father, the scholar, taught him, when the wise men talked. (Opening sentences)

Hermann Hesse

Siddhartha was Hesse’s ninth novel. Hesse had been born in 1877 into a devout Swabian Pietist household ‘with the Pietist tendency to insulate believers into small, deeply thoughtful groups’. He was an intensely serious young man who rebelled against his parents, tried to commit suicide, was sent to mental homes and then a boys’ institution, leaving school as soon as he could. He never attended university and became an apprentice at a bookshop. With few connections he struggled to get his early works of poetry or short fictions into print.

His breakthrough came with publication of the novel Peter Camenzind in 1904 and became popular throughout Germany. He married, had three children and supported himself for the rest of his life as a writer. Reading Schopenhauer had interested him in Eastern philosophy, and in the 1900s he read a lot about the subject.

Seven more novels followed. In 1911 he went on a trip to the East, to Sri Lanka, Borneo and Burma. On return it was clear his marriage was breaking down. The Great War broke out. His son fell ill and his wife developed schizophrenia. In 1916 Hesse went into psychotherapy, which led him to personal friendship with Freud’s disciple, Carl Jung. In 1919 Demian was published, then in 1922 Siddhartha.

The historical Buddha

The Buddha’s given name was Siddhārtha Gautama. He was born into an aristocratic family in what is present-day Nepal, around 480 BC (though his dates and all the facts relating to his life are open to extensive debate).

He renounced his privileged life and spent years travelling, learning, observing. One day he sat under the banyan tree and had a religious vision. He realised that all of life as commonly accepted amounts to duḥkha or suffering, and that only complete detachment from the wishes of the ego, the mind and body can bring complete detachment from self, and so achieve the end of dukkha – the state called Nibbāna or Nirvana.

‘Buddha’, by the way, is not a name but an adjective or title, meaning ‘Awakened One’ or the ‘Enlightened One’.

Siddhartha – part one

With fairy tale simplicity Hesse describes the efforts of Siddhartha, son of a worthy Brahmin in north India at the time of the Buddha, to attain wisdom. He meditates, he practices the ablutions and the rituals required of a high-caste Hindu Brahmin, and also reads the holy books, but he is discontent. He feels he will never attain wisdom this way.

And so he asks his father if he may leave in search of wisdom, Initially reluctant, his father lets him and, as he walks out of his ancestral village, Siddhartha is joined by his faithful friend, Govinda.

They spend ‘about three years’ (p.16) with the Samana, a sect of monks or spiritual devotees who live in the jungle, learning their ways. Then rumours arrive of a man named Gotama who is also known as the Buddha or enlightened one. Siddhartha asks the head Samana for permission to leave the community to go see this Gotama. This makes the head Samana angry, but Siddhartha (once again) overcomes all objections, and leaves.

Siddhartha and Govinda come to the town of Savathi, where Gotama has established a community of monks and followers, living in the Jetavana Grove just outside town, which a rich follower has given him.

In the morning they watch Gotama going to beg food for his mid-day meal, looking much like any other yellow-cloaked devotee. In the afternoon they hear him preach the four main points and the Eightfold Path, the way to escape the eternal recurrence of reincarnation into lives of suffering and pain, the way to escape from the cycle into the bliss of Nirvana.

Govinda is entranced and goes forward, with other pilgrims, to ask Gotama to take him into his community, and he is accepted. However, Siddhartha doesn’t. Siddhartha explains to Govinda that he has no doubt Gotama’s teachings are correct but he doesn’t wish to follow another man’s teachings, he wants to know.

Later he bumps into Gotama himself and politely asks permission to talk to him, and explains this conviction, that the Buddha’s teachings can be communicated and followed by others; but this isn’t what he’s after. He isn’t after teachings, the world is full of teachings. He is after the Buddha’s experience but that experience is, by definition, incommunicable.

Thus Siddhartha must leave the community and must find his own way. Gotama warns him against the chains of opinion and knowledge, and against being too clever.

‘Be on your guard against too much cleverness.’

But Siddhartha is determined and leaves the community, and his best friend Govinda behind.

Walking alone he has a revelation of his own – all this time, pursuing the teachings of the ancients or gurus, he has been motivated by one thing: fear of his Self, fleeing from his Self. What would happen if he accepted his own Self, his selfness, as supreme, as the basis of his existence.

‘I do not want to kill and dissect myself any longer, to find a secret behind the ruins. Neither Yoga-Veda shall teach me any more, nor Atharva-Veda, nor the ascetics, nor any kind of teachings. I want to learn from myself, want to be my student, want to get to know myself, the secret of Siddhartha.’

This is connected with a revelation of the multitudinousness of life, the blue sky and the green forest. Everything has a distinct itness. Trying to abolish the many in order to penetrate through to The One – as the Brahmins do – is a mistake.

Cleaving to his Self for the first time he feels genuinely alone, not a member of his caste or a pilgrim among pilgrims or a scholar among scholars. The world melts away and he stands like a star in the heavens. He is just Siddhartha, the one and only Siddhartha and the realisation makes ‘a feeling of icy despair’ go through him, but at the same time he is more awake than he’s ever been before. He is awakened. He is reborn.

Siddhartha – part two

Siddhartha walks through the world, enlightened. No longer does he reject and spurn the things of the world as a veil to be penetrated. The reverse: now he celebrates the amazing diversity, colour and beauty of the natural world.

But this second part is dominated by what happens next. Siddhartha takes a ferry over a river and comes to a town where he admires a beautiful woman being carried by four bearers on an ornamented sedan chair. He makes enquiries. It is Kamala the noted courtesan. He is struck. He goes into the town and has his beard cut off and his hair cut and oiled. He bathes in the river. Then he presents himself to Kamala’s people and she grants him an audience.

Long story short: he becomes her lover and best friend. She teaches him the forty ways of love, finding pleasure in every look, word and every part of the human body. She tells him she needs her lover to be rich and well-dressed and gives him an introduction to the town’s leading merchant, Kamaswami.

Siddhartha impresses Kamaswami with his education and calmness. He is hired into the business. He does well, but never really gains a taste for it, the business itself. Instead he brings calm, detachment, education and a winning manner which pleases clients.

The years pass. The awakening he experienced after leaving Gotama slowly fades. He acquires wealth, a house by the river, fine clothes. No longer a vegetarian, he eats meat, gets drunk on wine. His face grows lined and corrupt. He becomes addicted to gambling with dice, gambling for immense stakes, loses fortunes, wins back fortunes – all to show his contempt for ‘riches’ and all the things the little people value. His inner voice has grown silent. He is in his forties with his first grey hairs (p.65).

He goes to see Kamala and she, also, is upset. They make love deeply. He goes back to his house, feels sick and glutted, wishes he could vomit up his corrupt life. Goes into his pleasure garden, sits under his mango tree, reviews his life, thinks he has lost all the fire which motivated him to learn the Brahmin scriptures, to outdo Govinda in wisdom, everything he learned with the Samana and understood about the Buddha – and yet though he has gained the outer trappings of Kamaswami’s people, people of this world, he is not one of them. He is lower than them. They give themselves to their loves and passions and work and anxieties. Siddhartha only pretends, in this as in everything else.

He looks up at the stars above his mango tree and realises all this is dead to him. He says goodbye to his mango tree and his pleasure garden and his town house and walks away, leaving everything behind. Kamaswami sends out searchers but never hears of him, Kamala is saddened but gladdened that he has been true to himself. A few months later she realises she is pregnant with his child.

Siddhartha wanders. He comes to a river and is so overcome with disgust at what he has become that he leans over the river as if to fall in and drown. He is contemplating suicide. Then out of some remote part of his soul comes the word Om, the beginning and end of Brahmin prayers, the syllable of reality. And he stops, repeats the syllable, is suddenly overcome by tiredness, sinks down onto the roots of the tree and sleeps, the word Om echoing through his unconscious.

When he wakes he feels a new man, refreshed and cleansed. A monk is watching him. It is his old friend Govinda, who was passing with fellow Buddhist pilgrims and saw Siddhartha sleeping in this place which is dangerous for its snakes and wild animals, and decided to stop and look over him. Now he has awoken, Govinda will join his colleagues. Siddhartha says, Don’t you recognise me? The short answer is, No, because Siddhartha has become fat and lined and worn and is wearing rich man’s clothes. Siddhartha tells his old friend all of those attributes are fleeting. Beneath them all, he is still following his quest. Govinda digests this, then bows and goes his way.

Siddhartha reflects on how far astray his old life had led him. In fact he reviews his entire life and all its changes. He realised he was over-educated when he was young, fenced in with prayers and ablutions and meditation. He had to get out and experience the futility of riches and sensual love for himself. Now he knows. Now he has awoken refreshed, a new man, as if his long sleep was one long Om-based meditation.

It is the same river he was ferried across 20 years ago. It is the same ferryman who, after a bit of prompting, remembers him. Siddhartha says he wants to give the ferryman his fine clothes and in return become his apprentice. The ferryman’s name is Vasudeva. He accepts. Siddhartha moves in to share his humble house and food and learn the trade. Slowly the two men come to look alike, taking turns to ferry people across the wide river, or sitting in silence for hours listening to it, learning from its wisdom.

One day Siddhartha articulates to the ferryman what the river has taught him: it has surpassed Time. Its beginning, middle and end are all simultaneously present. It is always changing but always the same. Nothing is past or future, everything exists in a permanent present, including Siddhartha. The river is the voice of life, the voice of Being, of perpetual Becoming (p.87).

Then news comes. The Buddha is dying. The couple of old men find themselves ferrying increasing numbers of monks and pilgrims who want to see the Enlightened One before he attains Nirvana. Among them is Kamala who has long since abandoned her trade as courtesan, given her money and troth to the Buddha. Now she is travelling with her son by Siddhartha.

They stop to rest on the far side of the river and Kamala sleeps, but wakens with a cry. She has been bitten by a poisonous snake. Siddhartha and Vasudeva hasten to her side. They try to cleanse the wound but it is already turning black. Kamala is dying. She lingers long enough to recognise Siddhartha and say how pleased she is to see the old sparkle and happiness in his eyes. She proclaims the boy is his son. She had wanted to see the Enlightened One before she died, but is content to see Siddhartha, who has a wisdom of his own.

Kamala dies. They burn her body on a funeral pyre.

Soon Siddhartha realises that his 11-year-old son is a spoiled mummy’s boy. He thinks that by love and patience he can reconcile him to living with two ageing rice-eating poor men. But he can’t. The boy has tantrums, breaks things, is nothing but trouble.

One day Vasudeva takes him aside and tells him he must take the boy back to his own kind. There is a lesson here. Did not Siddhartha have to immerse himself in the destructive element of life, did it not take him decades to find his own path and his own wisdom? Well, he can’t short-circuit it for the boy. The boy should be returned to his own kind, to his mother’s house or to a teacher, to grow up among other rich children and find his own path.

But Siddhartha can’t bring himself to do it and the boy comes to hate him, defying him, speaking harsh words every day. Finally he steals their money, runs away, rows the ferry boat to the other side of the river and is gone. Vasudeva wisely counsels Siddhartha not to follow his errant son, but Siddhartha has to. The world and its pain are too much with him.

Siddhartha finds himself arriving at the edge of the town, by the old pleasure ground of Kamala. He stands transfixed, his mind full of memories of their young, ripe, hot-blooded time. He sits down in the dust, in a trance. He is only wakened when Vasudeva lightly touches his shoulder.

Back at the ferry, Siddhartha’s psychological wound – from the loss of his son – continues to chafe.

One day looking down into the river he realises his face reminds him of his father’s face, his father who he ran away from and never saw again and who probably died lonely, who probably suffered the same way Siddhartha is now suffering. How ridiculous, how absurd, the tragi-comic cycles of life, the endless repetition of suffering.

Vasudeva is getting old. He takes Siddhartha to sit by the river and listen. And Siddhartha hears all the voices of all the people, the plights, the lives as the river flows past, into the sea, evaporates into the sky, forms clouds over the hills, condenses and falls as rain which feeds a thousand springs which flow together to create the river. Eternal and ever-changing. And the thousands of voices converge to speak the syllable of perfection, Om.

Siddhartha feels healed, complete. He rises above his own personal suffering and becomes one with this vast unity of the world. And now Vasudeva stands and says it is time for him to slough off the skin of the ferryman Vasudeva and return to the unity of the cosmos. And he walks away from Siddhartha clothed in light.

In the final chapter Govinda arrives again. He had heard of a ferryman of great wisdom. Once again he doesn’t recognise Siddhartha till the latter announces himself. But the point of these last ten pages is that Govinda asks for help, for Siddhartha’s wisdom and when the latter explains it, it really is wisdom. It struck me with the force of a genuinely holy writing.

For Siddhartha explains that there is no such thing as time. All things are permanently present, all pasts and futures are contained in the now, and are part of a vast unity. If this is so then there are no real oppositions. Oppositions occur only in the words of teachings. To teach you have to take a view and be partial, separating x from y. But Siddhartha now scandalises Govinda by saying there is no real difference between Sansara, the Sanskrit word which betokens change and the eternal cycle of suffering, and Nirvana, the supposed heaven where the soul escapes the eternal cycle of suffering.

These, Siddhartha says, are just binary concepts required for clear doctrine and teaching. In reality everything is part of everything else. In this sense, there is no right or wrong, and certainly no good or bad. Good and bad are inextricably mixed, just as past and future are eternally present.

Therefore, the logical response, is to love the world as it is because it contains the entire future and all of heaven, here, now, implicitly. The correct attitude is complete compassion and complete love for everything as it is.

Govinda asks for a final word of help or advice and Siddhartha tells him to bend and kiss his forehead. And as he does so Govinda sees and hears all the voices of all the people in the world, all the babies, old people, lovers, warriors, priests and even gods and goddesses, a thousand thousand thousand voices and features, past and future, all contained in one vast cosmic unity. And he realises that only one other person has ever had the same level of wisdom and serenity and the same half-mocking smile on his lips. By a different route, Siddhartha has become as enlightened as the Buddha.

The personal quest

And so Siddhartha’s determination to go his own way is justified. The final wisdom, in practical terms, seems to be that everyone must find their own path:

There was no teaching a truly searching person, someone who truly wanted to find, could accept. But he who had found, he could approve of any teachings, every path, every goal, there was nothing standing between him and all the other thousand any more who lived in that what is eternal, who breathed what is divine.

Conclusion

This is a beautiful and inspiring book. You don’t necessarily have to agree with any of the Eastern philosophy on show, to find that many of the thoughts and ideas about life, about our paths through life, about trying to find meaning, ring a bell. Hesse’s novels have always been popular with the young, teenagers and students – but as a middle-aged parent I found much of what the characters discuss just as relevant to me, now, at this stage of my journey.

Above all, after over a thousand pages of bleakness, crudity, violence, rape, murder and madness in the novels of Hermann Broch and Alfred Döblin, it is a welcome relief to read a book in which people smile, enjoy the sight of the blue sky and the sound of a flowing river, are kind and wise and considerate and courteous to each other. It is like re-entering the real world after a prolonged visit to a lunatic asylum.

To put it another way, the longer Broch went on, the lengthier his dense and abstract and wordy philosophical disquisitions went on, the more impenetrable, hair-splitting, utterly academic and impractical they seemed. Whereas Hesse’s focused fable provides countless places where the character’s eloquent and strangely practical thoughts strike home to your heart and make you reflect on your own life and journey.


Related links

20th century German literature

  • The Tin Drum by Günter Grass (1959)

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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