The Art of the Novel by Milan Kundera (1986)

Need I stress that I intend no theoretical statement at all, and that the entire book is simply a practitioner’s confession? Every novelist’s work contains an implicit vision of the history of the novel, an idea of what the novel is; I have tried to express here the idea of the novel that is inherent in my own novels. (Preface)

This book contains seven essays on the art of the novel. First, a few observations.

Kundera is an academic Remember Kundera was a lecturer in ‘World Literature’ at Charles University in Prague for some 20 years (1952-75). This is a grand title and obviously encouraged a panoramic overview of the subject. Then he emigrated to France, where he continued to teach at university. He is, in other words, an academic, an expounder, a simplifier and teacher of other people’s views and theories, and that is probably the most dominant characteristic of his fiction – the wish to lecture and explicate.

He discusses a narrow academic canon You quickly realise he isn’t talking about the hundreds of thousands of novels which have been published over the past 400 years – he is talking about The Novel, the ‘serious novel’, ‘real novels’ – an entirely academic construct, which consists of a handful, well at most 50 novelists, across that entire period and all of Europe, whose concerns are ‘serious’ enough to be included in ‘serious’ academic study.

Non-British And he is very consciously European. This means many of his references are alien or exotic to us. Or just incomprehensible. When he says that The Good Soldier Schweik is probably the last popular novel, he might as well be living on Mars. There is no mention of Daniel Defoe, of Walter Scott, Jane Austen, Dickens, Trollope, George Eliot, Conrad, Henry James, DH Lawrence or Virginia Woolf, or anyone from the British ‘Great Tradition’ except the dry and dusty Samuel Richardson, in some histories, the founder of the English novel. He mentions Orwell’s ‘1984’ to dismiss it as a form of journalism. All Orwell’s fiction, he thinks, would have been better conveyed in pamphlets.

There is no mention of American fiction: from Melville through Twain, Hemingway and Faulkner (OK, Faulkner is mentioned right towards the end as one of the several authors who want nothing written about their lives, only their works), Updike or Roth or Bellow. No reference to science fiction or historical fiction or thrillers or detective fiction. Or children’s fiction. There is no mention of South American fiction (actually, he does mention a novel by Carlos Fuentes), or anything from Africa or Asia.

Some exceptions, but by and large, it is a very very very narrow definition of the Novel. Kundera can only talk as sweepingly as he does because he has disqualified 99.9% of the world from consideration before he begins.

1. The Depreciated Legacy of Cervantes (1983)

In 1935 Edmund Husserl gave a lecture titled ‘Philosophy and the Crisis of European Man’. He identifies the Modern Era as starting with Galileo (Dialogue Concerning the Two Chief World Systems, 1632) and Descartes (Discourse on the Method, 1637) and complains that Europe (by which he includes America and the other colonies) has become obsessed with science and the external world at the expense of spirit and psychology, at the expense of Lebenswelt.

Kundera says that Husserl neglected the novel, which was also born at the start of the modern era, specifically in the Don Quixote of Miguel Cervantes (1605). It is in the novel that Europeans have, for 400 years, been investigating the interior life of humanity. The novel discovers those elements of life which only it can discover. Therefore the sequence of great novelists amounts to a sequence of discoveries about human nature:

  • Cervantes – explores the nature of adventure
  • Richardson – the secret life of feelings
  • Balzac – man’s rootedness in history
  • Flaubert – details of the everyday
  • Tolstoy – the intrusion of the irrational into decision making
  • Proust – the elusiveness of time past
  • Joyce – the elusiveness of time present
  • Mann – the role of ancient myth in modern life

At the start of the Modern Era God began to disappear, and with him the idea of one truth. Instead the world disintegrated into multiple truths. In the novel these multiple truths are dramatised as characters.

The whole point of the novel is it does not rush to judgement, to praise or condemn. Religion and ideologies (and political correctness) does that. The whole point of the novel is to suspend humanity’s Gadarene rush to judge and condemn before understanding: to ‘tolerate the essential relativity of things human’ (p.7).

He describes how there is a straight decline in the European spirit, from Cervantes – whose heroes live on the open road with an infinite horizon and never-ending supply of adventures – through Balzac whose characters are bounded by the city, via Emma Bovary who is driven mad by boredom, down to Kafka, whose characters have no agency of their own, but exist solely as the function of bureaucratic mistakes. It’s a neat diagram, but to draw it you have to leave out of account most of the novels ever written – for example all the novels of adventure written in the later 19th century, all of Robert Louis Stevenson, for example.

As in all his Western books, Kundera laments the spirit of the age, how the mass media are making everything look and sound the same, reducing everything to stereotypes and soundbites, simplifying the world, creating ‘the endless babble of the graphomanics’ –  whereas the novel’s task is to revel in its oddity and complexity.

2. Dialogue on the Art of the Novel

In a written dialogue with an interviewer, Kundera moves the same brightly coloured counters around – Cervantes, Diderot, Flaubert, Proust, Joyce. The novel was about adventure, then about society, then about psychology.

He states his novels are outside the novel of psychology. There’s psychology in them but that’s not their primary interest.

Being a central European he sees the 1914-18 war as a catastrophe which plunged art and literature into the grip of a merciless History. The essential dreaminess of a Proust or Joyce became impossible. Kafka opened the door to a new way of being, as prostrate victim of an all-powerful bureaucracy.

He clarifies that a key concern is the instability of the self: which is why characters often play games, pose and dramatise themselves; it is to find out where their limits are.

He clarifies his approach as against Joyce’s. Joyce uses internal monologue. There is no internal monologue at all in Kundera. In fact, as he explains it, you realise that the monologue is his, the author’s as the author tries different approaches in order to analyse his own characters. His books are philosophical analyses of fictional characters. And the characters are conceived as ‘experimental selfs’ (p.31), fully in line with his core idea that the history of the novel is a sequence of discoveries.

If the novel is a method for grasping the self, first there was grasping through adventure and action (from Cervantes to Tolstoy). Then grasping the self through the interior life (Joyce, Proust). Kundera is about grasping the self though examining existential situations. He always begins with existential plights. A woman who has vertigo. A man who suffers because he feels his existence is too light, and so on. Then he creates characters around these fundamentals. Then he puts them into situations which he, the author, can analyse, analyse repeatedly and from different angles, in order to investigate the mystery of the self.

Thus a character is ‘not a simulation of a living being. It is an imaginary being. An experimental self.’ (p.34) Making a character ‘alive’ means getting to the bottom of their existential problem’ (p.35).

A novel examines not reality but existence. And existence is not what has occurred, existence is the realm of human possibilities, everything that man can become, everything he’s capable of. (p.42)

The novelist is neither historian nor prophet: he is an explorer of existence. (p.44)

The novel is a meditation on existence as seen through the medium of imaginary characters. (p.83)

A theme is an existential enquiry. (p.84)

3. Notes inspired ‘The Sleepwalkers’

The Sleepwalkers is the name given to a trilogy of novels by the Austrian novelist Hermann Broch (1886 – 1951). The three novels were published between 1928 and 1932. They focus on three protagonists and are set 15 years apart:

  1. Joachim von Pasenow set in 1888
  2. August Esch set in 1903
  3. Wilhelm Huguenau set in 1918

In their different ways they address on core them: man confronting the disintegration of his values.

According to Kundera, before one writes one must have an ontological hypothesis, a theory about what kind of world we live in. For example The Good Soldier Švejk finds everything about the world absurd. At the opposite pole, Kafka’s protagonists find everything about the world so oppressive that they lose their identities to it.

After all, What is action? How do we decide to do what we do? That is, according to Kundera, the eternal question of the novel. (p.58)

Through an analysis of the plots of the three novels, Kundera concludes that what Broch discovered was the system of symbolic thought which underlies all decisions, public or private.

He closes with some waspish criticism of ‘Establishment Modernism’, i.e. the modernism of academics, which requires an absolute break at the time of the Great War, and the notion that Joyce et al. definitively abolished the old-fashioned novel of character. Obviously Kundera disagrees. For him Broch (whose most famous masterpiece, The Death of Virgil didn’t come out till the end of World War Two) was still opening up new possibilities in the novel form, was still asking the same questions the novel has asked ever since Cervantes.

It is a little odd that Kundera takes this 2-page swipe at ‘Establishment Modernism’, given that a) he is an academic himself, and his own approach is open to all sorts of objections (mainly around its ferocious exclusivity), and b) as he was writing these essays, Modernism was being replaced, in literature and the academy, by Post-Modernism, with its much greater openness to all kinds of literary forms and genres.

4. Dialogue on the Art of Composition (1983)

Second part of the extended ‘dialogue’ whose first part was section two, above. Starts by examining three principles found in Kundera’s work:

1. Divestment, or ellipsis. He means getting straight to the heart of the matter, without the traditional fol-de-rol of setting scenes or background to cities or towns or locations.

2. Counterpoint or polyphony. Conventional novels have several storylines. Kundera is interested in the way completely distinct themes or ideas can be woven next to each other, setting each other off. For the early composers a principle of polyphony was that all the lines are clear and distinct and of equal value.

Interestingly, he chooses as fine examples of his attempts to apply this technique to his novels, the Angels section in The Book of Laughter and Forgetting – which I found scrappy and unconvincing – and Part Six of The Unbearable Lightness of Being, which I think is by far the worst thing he’s ever written, embarrassingly bad.

There’s some chat about Kundera’s own personal interventions in his novels. He emphasises that anything said within a novel is provisional hypothetical and playful. Sure, he intervenes sometimes to push the analysis of a character’s situation deeper than the character themselves could do it. But emphasises that even the most serious-sounding interventions are always playful. They can never be ‘philosophy’ because they don’t occur in a philosophical text.

From the very first word, my thoughts have a tone which is playful, ironic, provocative, experimental or enquiring. (p.80)

This is what he means by ‘a specifically novelistic essay’ i.e. you can write digressions and essays within novels but, by coming within its force field, they become playful and ironic.

The final part is an analysis of his novels in terms of their structure, their architecture i.e. the number of parts, the way the sub-sections are so distinct. And then a really intense comparison with works of classical music, in the sense that the varying length and tempo of the parts of his novels are directly compared with classical music, particularly to Beethoven quartets. Until the age of 25 he thought he was going to be a composer rather than a writer and he is formidably learned about classical music.

5. Somewhere behind (1979)

A short essay about Kafka. He uses the adjective Kafkan, which I don’t like; I prefer Kafkaesque. What does it consist of?

  1. boundless labyrinth
  2. a man’s life becomes a shadow of a truth held elsewhere (in the boundless bureaucracy), which tends to make his life’s meaning theological. Or pseudo-theological
  3. the punished seek the offence, want to find out what it is they have done
  4. when Kafka read the first chapter of The Trial to his friends everyone laughed including the author. Kafka takes us inside a joke which looks funny from the outside, but…

Fundamentally his stories are about the dehumanisation of the individual by faceless powers.

What strikes Kundera is that accurately predicted an entire aspect of man in the 20th century without trying to. All his friends were deeply political, avant-garde, communist etc, thought endlessly about the future society. But all of their works are lost. Kafka, in complete contrast, was a very private man, obsessed above all with his own personal life, with the domineering presence of his father and his tricky love life. With no thought of the future or society at large, he created works which turned out to be prophetic of the experience of all humanity in the 20th century and beyond.

This Kundera takes to be a prime example of the radical autonomy of the novel, whose practitioners are capable of finding and naming aspects of the existential potential of humanity, which no other science or discipline can.

6. Sixty-Three Words (1986)

As Kundera became famous, and his books published in foreign languages, he became appalled by the quality of the translations. (The English version of The Joke particularly traumatised him; the English publisher cut all the reflective passages, eliminated the musicological chapters, and changed the order of the parts! In the 1980s he decided to take some time out from writing and undertake a comprehensive review of all translations of his books with a view to producing definitive versions.

Specific words are more important to Kundera than other novelists because his novels are often highly philosophical. In fact, he boils it down: a novel is a meditation on certain themes; and these themes are expressed in words. Change the words, you screw up the meditations, you wreck the novel.

A friendly publisher, watching him slog away at this work for years, said, ‘Since you’re going over all your works with a fine toothcomb, why don’t you make a personal list of the words and ideas which mean most to you?’

And so he produced this very entertaining and easy-to-read collection of short articles, reflections and quotes relating to Milan Kundera’s keywords:

  • aphorism
  • beauty
  • being – friends advised him to remove ‘being’ from the title of The Unbearable Lightness of Being’: but it is designed to be a meditation on the existential quality of being. What if Shakespeare had written: To live or not to live… Too superficial. He was trying to get at the absolute root of our existence.
  • betrayal
  • border
  • Central Europe – the Counter-Reformation baroque dominated the area ensuring no Enlightenment, but on the other hand it was the epicentre of European classical music. Throughout the book he is struck by the way the great modern central European novelists – Kafka, Hasek, Musil, Broch, Gombrowicz – were anti-Romantic and modern just not in the way of the flashy avant-gardes of Rome or Paris. Then after 1945 central Europe was extinguished and – as he was writing this list – was a prophetic type of the extinguishment of all Europe. Now we know this didn’t happen.
  • collaborator – he says the word ‘collaborator’ was only coined in 1944, and immediately defined an entire attitude towards modernity. Nowadays he reviles collaborators with the mass media and advertising who he thinks are crushing humanity. (Looking it up I see the word ‘collaborator’ was first recorded in English in 1802. This is one of the many examples where Kundera pays great attention to a word and everything he says about it turns out to be untrue for English. It makes reading these essays, and his ovels, a sometimes slippery business.)
  • comic
  • Czechoslovakia – he never uses the word in his fiction, it is too young (the word and country were, after all, only created in 1918, after the Austro-Hungarian Empire collapsed). He always uses ‘Bohemia’ or ‘Moravia’.
  • definition
  • elitism – the Western world is being handed over to the control of a mass media elite. Every time I read his diatribes against the media, paparazzi and the intrusion into people’s private lives, I wonder what he makes of the Facebook and twitter age.
  • Europe – his books are streaked with cultural pessimism. Here is another example. He thinks Europe is over and European culture already lost. Well, that’s what every generation of intellectuals thinks. 40 years later Europe is still here.
  • excitement
  • fate
  • flow
  • forgetting – In my review of The Book of Laughter and Forgetting I pointed out that Mirek rails against forgetting as deployed by the state (sacking historians) but is himself actively engaged in trying to erase his past (claiming back his love letters to an old flame). Kundera confirms my perception. Totalitarian regimes want to control the past (‘Orwell’s famous theme’), but what his story shows is that so do people. It is a profound part of human nature.
  • graphomania – he rails against the way everyone is a writer nowadays, and says it has nothing to do with writing (i.e. the very careful consideration of form which he has shown us in the other essays in this book) but a primitive and crude will to impose your views on everyone else.
  • hat
  • hatstand
  • ideas – his despair at those who reduce works to ideas alone. No, it is how they are treated, and his sense of the complexity of treatment is brought out in the extended comparison of his novels to complicated late Beethoven string quartets in 4. Dialogue on the Art of Composition
  • idyll
  • imagination
  • inexperience – a working title for The Unbearable Lightness of Being was The Planet of Inexperience. Why? Because none of us have done this before. We’re all making it up as we go along. That’s what’s so terrifying, so vertiginous.
  • infantocracy
  • interview – as comes over in a scene in Immortality, he hates press interviews because the interviewer is only interested in their own agenda and in twisting and distorting the interviewees’ responses. Thus in 1985 he made a decision to give no more interviews and only allow his views to be published as dialogues which he had carefully gone over, refined and copyrighted. Hence parts two and four of this book, although they have a third party asking questions, are in the form of a dialogue and were carefully polished.
  • irony
  • kitsch – he’s obsessed with this idea which forms the core – is the theme being meditated on – in part six of the Unbearable Lightness of Being. It consists of two parts: step one is eliminating ‘shit’ from the world (he uses the word ‘shit’) in order to make it perfect and wonderful, as in Communist leaders taking a May Day parade or TV adverts. Step two is looking at this shallow, lying version of the world and bursting into tears at its beauty. Kitsch is ‘the need to gaze into the mirror of the beautifying lie and to be moved to tears of gratification at one’s own reflection.’ (p.135)
  • laughter – For Rabelais, the comic and the merry were one. Slowly literature became more serious, the eighteenth century preferring wit, the Romantics preferring passion, the nineteenth century preferring realism. Now ‘the European history of laughter is coming to an end’. (p.136) That is so preposterous a thought I laughed out loud.
  • letters
  • lightness
  • lyric
  • lyricism
  • macho
  • meditation – his cultural pessimism is revealed again when he claims that ‘to base a novel on sustained meditation goes against the spirit of the twentieth century, which no longer likes to think at all. (p.139)
  • message
  • misogynist – gynophobia (hatred of women) is a potential of human nature as is androphobia (hatred of men), but feminists have reduced misogyny to the status of an insult and thus closed off exploration of a part of human nature.
  • misomusist – someone who has no feel for art or literature or music and so wants to take their revenge on it
  • modern
  • nonbeing
  • nonthought – the media’s nonthought
  • novel and poetry – the greatest of the nivelists -become-poets are violently anti-lyrical: Flaubert, Joyce, Kafka (don’t think that’s true of Joyce whose prose is trmeendously lyrical)
  • novel – the European novel
  • novelist and writer
  • novelist and his life – quotes from a series of novelists all wishing their lives to remain secret and obscure: all attention should be on the works. Despite this, the army of biographers swells daily. The moment Kafka attracts more attention that Josef K, cultural death begins.
  • obscenity
  • Octavio – the Mexican writer, Octavio Paz
  • old age – frees you to do and say what you want.
  • opus
  • repetitions
  • rewriting – for the mass media, is desecration. ‘Death to all those who dare rewrite what has been written!’ Jacques and His Master
  • rhythm – the amazing subtlety of rhythm in classical music compared to the tedious primitivism of rock music. Tut tut.
  • Soviet – the Germans and Poles have produced writers who lament the German and Polish spirit. The Russians will never do that. They can’t. Every single one of them is a Russian chauvinist.
  • Temps Modernes – his cultural pessimism blooms: ‘we are living at the end of the Modern Era; the end of art as conceived as an irreplaceable expression of personal originality; the end that heralds an era of unparalleled uniformity’ (p.150)
  • transparency – the word and concept in whose name the mass media are destroying privacy
  • ugly
  • uniform
  • value – ‘To examine a value means: to try to demaracte and give name to the discoveries, the innovations, the new light that a work casts on the human world.’ (p.152)
  • vulgarity
  • work
  • youth

7. Jerusalem Address: the Novel and Europe (1985)

In the Spring of 1985 Kundera was awarded the Jerusalem Prize. He went to Jerusalem to deliver this thank you address. It is a short, extremely punch defense of the novel as a form devoted to saving the human spirit of enquiry in dark times.

In a whistlestop overview of European history, he asserts that the novel was born at the birth of the modern era when, with religious belief receding, man for the first time grasped his plight as a being abandoned on earth: the novel was an investigation of this plight and has remained so ever since.

The novel is the imaginary paradise of individuals. It is the territory where no one possesses the truth… but where everyone has the right to be understood. (p.159)

Every novel, like it or not, offers some answer to the question: What is human existence, and wherein does its poetry lie? (p.161)

But the novel, like the life of the mind, has its enemies. Namely the producers of kitsch and what Rabelais called the agélastes, people who have no sense of humour and do not laugh. He doesn’t say it but I interpret this to mean those who espouse identity politics and political correctness. Thou Must Not Laugh At These Serious Subjects, say the politically correct, and then reel off a list which suits themselves. And kitsch:

Kitsch is the translation of the stupidity of received ideas into the language of beauty and feeling. It moves us to tears of compassion for the banality of what we think and feel. (p.163)

The greatest promoter of kitsch is the mass media which turns the huge human variety into half a dozen set narratives designed to make us burst into tears. We are confronted by a three-headed monster: the agélastes, the nonthought of received ideas, and kitsch.

Kundera sees European culture as being under threat from these three forces, and identifies what is most precious about it (European culture), namely:

  • its respect for the individual
  • for the individual’s original thought
  • for the right of the individual to a private life

Against the three-headed monster, and defending these precious freedoms, is set the Novel, a sustained investigation by some of the greatest minds, into all aspects of human existence, the human predicament, into human life and interactions, into human culture.


Central ideas

The novel is an investigation into man’s Lebenwelt – his life-being.

Novelists are discoverers and explorer of the capabilities, the potentialities, of human existence.

Conclusions

1. Fascinating conception of the novel as a sustained investigation into the nature of the self, conducted through a series of historical eras each with a corresponding focus and interest.

2. Fascinating trot through the history of the European novel, specially the way it mentions novelists we in England are not so familiar with, such as Hermann Broch or Diderot or Novalis, or gives a mid-European interpretation to those we have heard of like Kafka or Joyce.

3. Fascinating insight into not only his own working practice, but what he thinks he’s doing; how he sees his novels continuing and furthering the never-ending quest of discovery which he sees as the novel’s historic mission.

But what none of this fancy talk brings out at all, is the way Milan Kundera’s novels are obsessed with sex. It is extraordinary that neither Sex nor Eroticism appear in his list of 63 words since his powerfully erotic (and shameful and traumatic and mysterious and ironic) explorations of human sexuality are what many people associate Kundera’s novels with.

Last thoughts

Changes your perspective It’s a short book, only 165 pages with big gaps between the sections, but it does a very good job of explaining how Kundera sees the history and function of the novel, as an investigation into the existential plight of humanity. It changed my mental image of Kundera from being an erotic novelist to being more like an existentialist thinker-cum-writer in the tradition of Sartre.

The gap between Britain and Europe There is a subtler takeaway, which is to bring out how very different we, the British, are from the Europeans. True, he mentions a few of our authors – the eighteenth century trio of Richardson, Fielding and Sterne – but no Defoe, Austen, Scott or Dickens.

The real point is that he assumes all European intellectuals will have read widely in European literature – from Dante and Boccaccio through Cervantes and into the eighteenth century of Diderot, Voltaire, the Marquis de Sade. And when you read the French founders of critical theory, Barthes or Derrida, or the influential historian Foucault, they obviously refer to this tradition.

But it remains completely alien to us in Britain. Not many of us read Diderot or Novalis or Lermontov or even Goethe. We’ve all heard of Flaubert and Baudelaire because, in fact, they’re relatively easy to read – but not many of us have read Broch or Musil, and certainly not Gombrowicz. Though all literature students should have heard of Thomas Mann I wonder how many have read any of his novels.

My point being that, as you read on into the book, you become aware of the gulf between this huge reservoir of writers, novels and texts in the European languages – French, German and Russian – and the almost oppressively Anglo-Saxon cultural world we inhabit, not only packed with Shakespeare and Dickens, but also drenched in American writers, not least the shibboleths of modern American identity politics such as Toni Morrison or Maya Angelou.

Reading this book fills your mind with ideas about the European tradition. But at the same time it makes you aware of how very different and apart we, in Britain, are, from that tradition. Some of us may have read some of it; but none of us, I think, can claim to be of it.

Credit

The Art of the Novel by Milan Kundera was first published in French in 1986. The English translation was published by Grove Press in the USA and Faber and Faber in the UK in 1988. All references are to the 1990 Faber paperback edition.


Related links

Milan Kundera’s books

1967 The Joke
1969 Life Is Elsewhere
1969 Laughable Loves (short stories)

1972 The Farewell Party
1978 The Book of Laughter and Forgetting

1984 The Unbearable Lightness of Being
1986 The Art of the Novel (essays)

1990 Immortality
1995 Slowness
1998 Identity

2000 Ignorance
2014 The Festival of Insignificance

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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