All posts tagged Emmanuel Swedenborg

A Universal History of Infamy by Jorge Luis Borges (1935, revd. 1954)

The book is no more than appearance, than a surface of images; for that very reason, it may prove enjoyable.
(Borges’s 1954 preface to A Universal History of Infamy)

Long ago

One thinks of Borges as a modern classic so it comes as a bit of a surprise to learn just how long ago he was writing. Born in 1899, Borges published his first book in 1923 and wrote steadily for the next 60 years (he died in 1986). In his long life he published an enormous number of volumes (‘In addition to short stories for which he is most noted, Borges also wrote poetry, essays, screenplays, literary criticism, and edited numerous anthologies’) and the Wikipedia bibliography lists 66 volumes of prose, poetry and essays, in total.

Which makes it all the more odd or unfair that he is still best known in the English-speaking world for more or less one volume, Labyrinths, and a handful of lesser works. Borges had published the following before we get to the book under review:

Fervor de Buenos Aires (1923) poetry
Inquisiciones (1925) essays
Luna de Enfrente (1925) poetry
El tamaño de mi esperanza (1925) essays
El idioma de los argentinos (1928) essays
Cuaderno San Martín (1929) poetry
Evaristo Carriego (1930) essays
Discusión (1932) essays

You’d expect poetry from a starter author, but it’s notable that so many of these early volumes contain essays, in other words short prose explorations of ideas – about other authors, historical events or topics etc. It was for his short essays on imaginary or fantastical subjects that he was to become famous and A Universal History of Infamy, more or less the earliest work by Borges you can read in English translation, gives an indication why.

A Universal History of Infamy

A Universal History of Infamy is not, in fact, a universal history of infamy or anything like that ambitious. In reality it is much smaller in scope, and consists of:

seven ‘biographical essays’ – witty, ironic accounts of legendary bad guys and women from history whose stories Borges has cherry picked from his highly eclectic reading
one relatively straightforward short piece of fiction
eight summaries of stories or anecdotes he had come across in arcane sources and which attracted Borges for their fantastical or humorous aspects

Most of the essays had been published individually in the Argentine newspaper Crítica between 1933 and 1934. The 1934 collection was revised and three new stories added in the 1954 edition. There are two English translations of the book. The one I own dates from 1972 and was translated by Borges’s long-standing English translator, Norman Thomas di Giovanni. The 2004 English edition gives the stories slightly different titles.

The title A Universal History of Infamy derives from the fact that the seven biographical essays are fictionalised accounts of real-life criminals. The textual sources for each biography are listed at the end of the book: for example, the essay about the Widow Ching cites a 1932 History of Piracy as its source, the essay on Monk Eastman cites Herbert Asbury’s 1928 history of The Gangs of New York, the essay about Lazarus Morell cites Mark Twain’s Life on the Mississippi, the one about Tom Castro cites the Encyclopedia Britannica as its source, and so on.

So the sources are a) not particularly recondite and b) they were often fairly recent to Borges’s time of writing, in some cases published only a year or so before Borges wrote his potted summaries.

That said, Borges treats his sources very freely, changing dates, incidents and even names as he fancied to make his fantasy biographies deliberately fanciful and untrustworthy.

Part 1. Seven infamy stories

So these are stories Borges found in other books during his wide and eclectic reading and which attracted him for their elements of the macabre or gruesome, and which he chose to retell, dropping or adding details as he saw fit.

The Dread Redeemer

Lazarus Morell is poor white trash who grew up on the banks of the Mississippi and as an adult comes to be a leader of crooks who devise the following scam: they persuade gullible black slaves to run away from their owners and allow themselves to be sold on by the Morell gang who promise to liberate them and share the proceeds of this sale. But they don’t. They have ‘liberated’ some 70 slaves in this manner until the gang is joined by Virgil Stewart, famous for his cruelty, who promptly betrays them to the authorities. Morell goes into hiding in a boarding house, then, after 5 days, shaves off his beard and makes an escape to round up what remains of his gang and try to create a mass uprising of the southern slaves and lead a takeover of the city of New Orleans. Instead he dies of a lung ailment in Natchez hospital in January 1835 under an assumed name.

This story is quite florid enough to satisfy anyone’s taste for the lurid and melodramatic. What tames and raises it from being a shilling shocker is Borges’s dry wit and irony.

Morell leading rebellions of blacks who dreamed of lynching him; Morell lynched by armies of blacks he dreamed of leading – it hurts me to confess that Mississippi history took advantage of neither of these splendid opportunities. Nor, contrary to all poetic justice (or poetic symmetry), did the river of his crimes become his grave.

We expected a grand finale? Sorry folks.

If Borges’s narrative ends playfully, it opens even more so, with Borges referencing Spanish missionary Bartolomé de las Casas. Why? Because it was de las Casas who (apparently) had the bright idea of importing African slaves to work the silver mines of the newly discovered New World. Borges phrases this with characteristic irony (or is it facetiousness?)

In 1517, the Spanish missionary Bartolomé de las Casas, taking great pity on the Indians who were languishing in the hellish workpits of Antillean gold mines, suggested to Charles V, king of Spain, a scheme for importing blacks, so that they might languish in the hellish workpits of Antillean gold mines.

That is an example of what you could call literal facetiousness, the repetition of the initial heartless description being so unexpected as to be funny. But it then expands into a more grandiose type of joke as Borges goes on to deliver an unexpected perspective on the results of de las Casas’ brainwave i.e. the vast and numerous consequences of the invention of African slavery, and proceeds to a mock encyclopedic list of some of its untold consequences, namely:

W. C. Handy’s blues; the Parisian success of the Uruguayan lawyer and painter of Negro genre, don Pedro Figari; the solid native prose of another Uruguayan, don Vicente Rossi, who traced the origin of the tango to Negroes; the mythological dimensions of Abraham Lincoln; the five hundred thousand dead of the Civil War and its three thousand three hundred millions spent in military pensions; the entrance of the verb ‘to lynch’ into the thirteenth edition of the dictionary of the Spanish Academy; King Vidor’s impetuous film Hallelujah; the lusty bayonet charge led by the Argentine captain Miguel Soler, at the head of his famous regiment of ‘Mulattoes and Blacks’, in the Uruguayan battle of Cerrito; the Negro killed by Martín Fierro; the deplorable Cuban rumba ‘The Peanut Vender’; the arrested, dungeon-ridden Napoleonism of Toussaint L’Ouverture; the cross and the snake of Haitian voodoo rites and the blood of goats whose throats were slit by the papaloi’s machete; the habanera, mother of the tango; another old Negro dance, of Buenos Aires and Montevideo, the candombe.

The intellectual pleasure derives from the combination of mock scholarliness with the pleasing randomness of the examples selected. And not only surreal but – and this is an important part of Borges’s appeal – conveying an enormous sense of spaciousness; the sense of an enormously well-read mind, overflowing with wonderful facts and references, from the obvious to the fantastically recondite and abstruse. And that by reading along with Borges, we too, become as fantastically learned and knowledgeable as him.

If you like this kind of subject matter, and the dry ironical tone, then the world of unexpected and outré references is like a door opening in your mind, hundreds of doors, revealing all kinds of wonderful, mind and spirit enhancing vistas and possibilities.

Tom Castro, the Implausible Imposter

Arthur Orton was born in Wapping in 1834. He ran away to sea and resurfaced decades later in Sydney Australia where he had taken the name Tom Castro. Here he became friendly with a stately, clever black man, Ebenezer Bogle and the two set up as con-men. In 1854 a British steamer sank in the Atlantic and one of the passengers lost was slender, elegant Roger Charles Tichborne, heir to one of the greatest Roman Catholic families in England. His mother, Lady Tichborne, refused to believe he was dead and advertised widely throughout the colonies for his return. With wild and hilarious improbability Orton and Bogle decide to reply to her and claim that obese illiterate Tom Castro is in fact her slender, elegant aristocratic son…after some years of living in Australia!

Most of this is comic but Borges milks it for further comic ideas, such as the notion that it was the very outrageousness of the entire idea which gave Bogle and Orton confidence; the more ridiculous it seemed, the more emboldened they were to tough it out in the light of lawyers and Lady Tichborne’s heirs who violently rejected their claim. Very funny is the notion that so mad is Lady Tichborne to have her son restored that she will accept anything Orton says and so when he completely invents some tender childhood memories, Lady T immediately accepts them and makes them her own.

Finally, the relatives bring a trial where all is going well until Bogle meets his death at the hands of a passing hansom cab and Orton loses all his confidence. He is sentenced to 14 years in prison but, here again Borges emphasises the humour, pointing out that Orton so charmed his imprisoners that he was let off for good behaviour and then took to touring theatres giving a one-man show retelling his story.

It is typically Borgesian that, at each venue, Orton is described as starting out maintaining his innocence but often ends up pleading guilty depending on the mood of the audience.

A story, any story, about anything, is infinitely malleable.

The Widow Ching, Lady Pirate

China at the turn of the 18th century and the story of a redoubtable woman pirate who, when her husband Ching is killed in battle, takes over his pirate crew and leads them in 13 years of ‘systematic adventure’. The emperor sends one admiral against her, Admiral Kwo-lang, who she comprehensively defeats, and before leading her ‘six hundred war junks and forty thousand victorious pirates’ on devastating attacks on China’s seaboards. A second expedition is sent under one Ting-kwei. This one defeats Madame Ching who, on the night after a huge and bloody battle, has herself rowed over to the admiral’s ship, boards it and presents herself with the appropriately flowery oriental rhetoric: ‘the fox seeks the dragon’s wing.’ She was allowed to live and devoted her later years to the opium trade.

There is something immensely satisfying in the way Borges creates a scene, a historical period, its key characters and conveys a series of big events in just nine pages. More than that, the first page is devoted to two women pirates of the Western tradition, Mary Read and Anne Bonney, before we even get round to China.

Their speed and brevity, their exotic setting and subject matter, the tremendous confidence with which Borges cuts from scene to scene, zeroing in on key moments and drilling down to one line of dialogue, and all told in a wonderfully humorous, often tongue-in-cheek style, make these bonne bouches immensely appetising and pleasurable.

Monk Eastman, Purveyor of Iniquities

Borges freely acknowledges his source for this narrative as Herbert Asbury’s 1928 volume The Gangs of New York, and gives a 2-page summary of some of the most notable hoodlums from New York’s Victorian underworld described in that book, before arriving at his potted biography of ‘Monk’ Eastman who is the particular subject of this narrative.

Born Edward Osterman, Eastman he was Jewish but grew into a ‘colossal’ and violent killer who lorded it over the whole goy underworld. He hired himself out as a hitman and led a violent gang. They were involved in a shootout so epic it became known as The Battle of Rivington Street, then a two-hour fistfight with the leader of the main rival gang, Paul Kelly, watched by a shouting crowd. He was repeatedly arrested and, after the final time, in 1917, decided to enlist in the US Army which had joined the war in Europe. This, like everything else in the story, is told with detached facetiousness:

We know that he violently disapproved of taking prisoners and that he once (with just his rifle butt) interfered with that deplorable practice…

On his return Monk quipped that ‘a number of little dance halls around the Bowery were a lot tougher than the war in Europe.’ He was found dead in an alley with five bullets in him. These throwaway endings, without any Victorian moralising, give them a Modernist, ‘so what’ aspect, a throwaway bluntness which contrasts vividly with the extreme scholarly punctiliousness about the sources.

The Disinterested Killer Bill Harrigan

Scenes from the life of William Harrigan aka Billy the Kid. For a start, it’s factually interesting to learn that Billy was a street hoodlum born in the very tough slums of New York before he headed out West. Borges amuses himself by assigning Billy’s life to different stages, namely:

The larval stage
Go West!
The Demolition of a Mexican
Deaths for Deaths’ Sake

He killed his first man aged 14. There’s a running joke that whenever Billy boasted about the number of men he killed he always added ‘not counting Mexicans’ who he held in utter contempt.

Borges’s wonderful fantasy-mindedness, the way he can introduce a mind-teasing idea into even the most obviously material occurs when he casually mentions that, despite his best efforts to turn himself into a hard-riding cowboy, Billy:

never completely matched his legend, but he kept getting closer and closer to it.

This implication that the legend of Billy the Kid existed before he began enacting it, and that he was fated to aspire to match his own legend… there is something wonderfully dizzying about this metaphysical-magical perspective, a dizzying magic metaphysical worldview which was to emerge more powerfully in his famous mid-career stories and excerpts.

The Insulting Master of Etiquette Kôtsuké no Suké

To be honest I didn’t understand this one, even after reading it twice. It’s set in Japan in 1702. An imperial envoy comes to stay with Asano Takumi no Kami who has been ‘trained’ by a rude and dismissive master of etiquette, Kira Kôtsuké no Suké. Asano was rude to the imperial envoy who, as a result, had him executed. Asano’s other retainers came to Kira Kôtsuké no Suké and told him, that since the error stemmed from his poor training of Asano, he should commit hara-kiri, but he refused and ran away and barricaded himself into a palace. Asano’s 47 retainers laid siege to the palace, broke in, discovered he had hidden, found him and killed him. That is why the story is sometimes called ‘The Learned History of the Forty-seven Retainers.

The Masked Dyer, Hakim of Merv

The story of Hakim, born in 736, who grows up to assume the identity of the Prophet of the Veil and establish a religion to rival Mohammed’s, by telling the impressionable that a messenger from God had come down from heaven, cut off his head and carried it up to heaven to receive a divine mission from Allah. He crystallises his position when, amid a crowded caravan, someone releases a leopard which Hakim appears to quell with the power of his eyes alone. He becomes the Veiled Prophet or Masked One and leads his followers to military victory, taking cities. He keeps a harem of one hundred and fourteen blind women.

He promulgated a belief system derived from the Christian Gnostics, namely that the world is a parody of Divine Reality, created by nine emanations from the original.

The world we live in is a mistake, a clumsy parody. Mirrors and fatherhood, because they multiply and confirm the parody, are abominations.

Five years into his rule, Hakim and his followers are besieged by the army of the Caliph when a rumour goes round from one of the women of his harem that his body has various imperfections. He is praying at a high altar when two of his captains tear away his permanent veil to reveal that Hakim bears the revolting disfigurements of the leper, and they promptly run him through with spears.

Part 2. A short story

Man on Pink Corner

This is a surprisingly poor short story and a good explanation of why Borges focused on writing his metaphysical-brainteasing essays rather than trying any attempt at conventional fiction. It’s the account of a street hoodlum, a junior member of a gang in the unfashionable poor north side of Buenos Aires, and a supposedly fateful night when he and his gang are at a dance hall when in crashes a massive hard man, Francisco Real, who muscles his way through the crowd to confront the head of the local gang, Rosendo Juárez, at which point, inexplicably, Rosendo backs down and Real takes his place as head honcho and steals his woman, La Lujanera.

I found a lot of this inconsequential, silly and hard to follow because nobody seemed to be obeying any rules of human nature I’m familiar with. Rosendo disappears, and Real takes La Lujanera outside, presumably to copulate with her in a ‘ditch’:

By then they were probably going at it in some ditch.

Our narrator wanders out to take the air then returns to the dance where old gang members and new gang members seem to be dancing happily. Then there’s a banging on the door, and in stumbles the huge bruiser Real with a big gash in his chest. He collapses on the floor and bleeds to death. The hoodlums of both gangs strip him of his clothes, appear to rip open his guts and pull out his intestines, cut off his finger to steal his ring, then chuck him out the window into the river Maldonado which flows just outside the building.

In the final paragraph, the narrator mentions Borges’s own name as if he is recounting this story directly to him:

Then, Borges, I put my hand inside my vest – here by the left armpit, where I always carry it – and took my knife out again…

And in the last sentence implies that it was he, the narrator who, when he slipped out, managed to fatally stab Real – in which case why wasn’t there a description of this presumably fairly melodramatic scene, how did he manage to do it if Real was shagging La Lujanera in a ditch? How come La Lunajera didn’t point out our narrator to everyone in the hall as the murderer?

It seemed to me a collection of 1930s noir crime, lowlife clichés thrown together with no plausibility and no account of human psychology. Borges himself seemed bemused by the story’s popularity. Thank God he abandoned this mode of altogether in favour of the ‘baroque’ and mind-bending essays gathered in Labyrinths.

Part 3. Etcetera Etcetera

Being short 2 or 3-page excerpts from scholarly books which presumably struck Borges because of the surrealism or bizarreness or humour of their content. The excerpts are interesting or amusing or ghoulish in their own right, but what really impresses is the arcane nature of their sources, and the range of reading and learning they imply.

A Theologian in Death

From the Arcana Coelestis by Emanuel Swedenborg (1749 to 1756).

The Protestant theologian Philip Melancthon (1497 to 1560) dies and goes to heaven but doesn’t realise this is what has happened because the angels recreate his worldly house and study. However, as a great and learned man, they pester him to write about charity. But Melancthon obstinately persists in writing that charity is unnecessary because, like a zealous Protestant, he believes we are justified by faith alone. The result is that the angels , as warning and punishment, slowly degrade his house and then his own body. Day by day ghost Melancthon awakes in a further degenerated condition, till the last that’s heard of him he is ‘a kind of servant to demons.’

The Chamber of Statues

From The Thousand and One Nights numbers 271 and 272.

In the Andalucian city of Ceuta was a citadel with a door to which each successive king by tradition added a lock. Then a wicked man usurped the throne and, against the advice of holy men, insisted on ripping out the locks and opening the door to find what was inside. He discovered a series of rooms containing wonders, the last of which contained an inscription saying whoever opened the door would be overthrown. And, indeed, within a twelvemonth, the Arab leader Tariq ibn-Ziyad overthrew the usurper and sold his women and children into slavery.

Tale of the Two Dreamers

From The Thousand and One Nights number 351.

A merchant in Cairo falls asleep in his garden with a fountain and a fig tree and has a dream in which angels tell him to seek his fortune in Isfahan in Persia. So he packs up and off he sets.

After a gruelling journey, overcoming numerous threats and natural disasters, he finally arrives in Isfahan and falls asleep by a mosque.

But that night a house next to the mosque is robbed, the owners raise the alarm, the stranger is apprehended, thrown into prison and tortured. He is brought before the captain who asks who he is and why he’s here.

The merchant tells the story of his dream, and the captain laughs and says he also has a dream of a garden of a house in Cairo with a fig tree and a fountain which has treasure buried under it, but he knows it’s just a dream and has never acted on it.

He lets the whipped merchant go, who returns all the way back to his house in Cairo, digs under the fountain, and discovers a vast treasure. So the dream came true, just not at all in the way expected.

The Wizard Postponed

From the Libro delos enxiemplos del Conde Lucanor at de Patronio (1335) by Juan Manuel.

A beguiling story in which a Dean from Santiago, wanting to learn about magic, visits the noted magician Don Illán of Toledo and promises him anything if he will teach him magic.

So the Don takes him down into a cellar deep underground and submits him to a test, namely telescoping the next thirty years of their lives together. In this quick journey through the future the Dean is blessed with a series of promotions within the Catholic church, ending up being elected Pope.

At each step of the way the Don asks for some grace or favour but the newly promoted Dean puts him off, until he finally gets fed up of him and tells the Don to stop bothering him or he’ll have him thrown in prison.

At which point the entire future they’ve lived through disappears in a puff of smoke and the Dean finds himself back in the deep cellar with Don Illán who says ‘I told you so’, escorts him to the door and wishes him a pleasant journey home.

The Mirror of Ink

From The Lake Regions of Central Africa (1860) by Richard Burton.

How the wizard Abd-er-Rahman al-Masmudi threw himself on the mercy of the tyrant of Sudan Yaqub the Ailing, who orders him every morning to show him visions and wonders, until one day al-Masmudi shows him a figure being dragged for execution. When Yaqub demands that the figure’s veil be taken off, it reveals his own face and he watches the executioner raise his great sword and, when it falls and severs the neck of the man in the vision, Yaqub the Ailing himself falls dead.

A Double for Mohammed

From Vera Cristiana Religio (1771) by Emmanuel Swedenborg.

Since the idea of Mohammed is so closely linked to religion in the minds of Muslims, Allah ensures that heaven is overseen by a kind of deputy or second Mohammed, whose identity actually varies. A community of Muslims was once incited by evil spirits to acclaim Mohammed as their God, so Allah brought the spirit of the actual Mohammed up from under the earth to instruct them.

The Generous Enemy

From the Anhang zur Heimskringla (1893) by H. Gering

In 1102 Magnus Barfod undertook to conquer Ireland. Muirchertach, King of Dublin, sends him a nine-line curse which, by roundabout means, ends up coming true.

On Exactitude in Science

From Travels of Praiseworthy Men (1658) by J.A. Suárez Miranda.

A fragment which tells of a magical empire where the geographers at first essayed maps so huge that the map of a single province covered the space of an entire city, and the map of the Empire itself an entire Province. These were eventually replaced by the ultimate map of the empire which was the same size as the Empire itself, and coincided with it point for point. Over the years it fell into neglect and now only a few tattered fragments survive in the Western Deserts, sheltering an occasional beast or beggar.

Borges’ approach

Bookish

The content of the seven infamy tales is lurid and melodramatic, with plenty of murders, assassinations, beheadings, shootouts and suicides. But they are all refracted through a highly bookish, ironic sensibility which does at least two things: 1. It is very careful to cite the sources of the story, in a parody of a learned or scholarly article, and 2. it mocks the content of his own story with irony and knowing humour.

The first quality (a showy, pseudo-academic concern with indicating sources) is most evident in the opening of The Masked Dyer, Hakim of Merv:

If I am not mistaken the chief sources of information concerning Mokanna, the Veiled (or, literally, Masked) Prophet of Khurasan, are only four in number: a) those passages from The History of the Caliphs culled by Baladhuri; b) The Giant’s Handbook, or Book of Precision and Revision, by the official historian of the Abbasids, Ibn abi Tahir Taifur; c) the Arabic codex entitled The Annihilation of the Rose, wherein we find a refutation of the abominable heresies of the Dark Rose, or Hidden Rose, which was the Prophet’s Holy Book; and d) some barely legible coins unearthed by the engineer Andrusov during excavations for the Trans-Caspian railway. (p.77)

‘If I am not mistaken’, that’s a nice touch. The effect of these kinds of learnèd references is to give the very pleasurable sense that you are entering the magical realm of books and stories. Not the everyday books we encounter in our lives or local bookshops, glossy gardening books or biographies of celebrity chefs or tedious accounts of adulteries in North London – but that we have been transported to the realm of old-fashioned stories, stories of extreme actions and derring-do and marvellous deeds in exotic settings.

Stories from our remembered childhood which fired our imaginations before we were forced to grow up and become sensible. It is a very old-fashioned tone and it’s no surprise that Borges, throughout his career, said his earliest and most enduring inspiration derived from the yarns of Robert Louis Stevenson and Arthur Conan Doyle.

Exotic

This old-fashioned, bookish tone overlaps with the wonderfully exotic settings of many of the narratives: slave plantations of the Deep South; Australia; the China seas; 18th century Japan; the Wild West; 12th century Ireland; medieval Spain; medieval Persia.

In the first preface he mentions Robert Louis Stevenson as a source and you can feel Stevenson’s restless quest for exotic locations shared by Borges.

Intellectual themes

One of the most obvious recurring tropes of the stories is (the currently very modish theme of) ‘identity’. The seven historical characters freely change their names or have names assigned them by contemporaries or historians. Writing of Monk Eastman, he says:

These shifts of identity (as distressing as a masquerade, in which one is not quite certain who is who) omit his real name – presuming there is such a thing as a real name.

The most flagrant example is Tom Castro who has already changed his name once before he embarks on the criminal project of impersonating Roger Charles Tichborne, which leads to the sensational trial in which the nature of ‘identity’ is central. A great deal could be said on the subject of fiction and identity but I’m going to pass.

Two prefaces

After the fact, Borges commented on his own stories in two prefaces, one written for the 1934 edition, one for 1954.

1934 preface

The 1934 preface is only one page long and Borges admits that the stories stem, in part, from:

my rereadings of Stevenson and Chesterton, and also from Sternberg’s early films, and perhaps from a certain biography of Evaristo Carriego

combined with the over-use of certain tricks:

random enumerations, sudden shifts of continuity, and the paring down of a man’s whole life to two or three scenes

I found it very interesting indeed that he casually says:

They are not, they do not try to be, psychological.

Traditional literature, and many short stories, focus on a psychological crux, a decisive moment in someone’s life, and investigate the ‘moral’ and psychological aspects of it. Borges consciously turns his back on that tradition and exploits his sources to create pen portraits which are not at all concerned with anyone’s inner life, but use the content as 1. entertainment, creating striking scenarios and tableaux, as if in paintings or – as he frequently remarks – like scenes from movies. In the 1954 preface he elaborates that:

The book is no more than appearance, than a surface of images; for that very reason, it may prove enjoyable

They are intended to be all surface. That partly explains why they end so abruptly and with no moralising whatsoever: to emphasise their shiny metallic surfaceness.

2. What Borges doesn’t mention is that the stories are also quite clearly used as starting points for ironic and amused meditations on ideas, the more metaphysical and paradoxical the better. And that this was a harbinger of the work which was to come later.

1954 preface

The 1954 preface is twice as long as the 1934 one, being an extravagant 2 pages in length. Borges immediately launches into a consideration of ‘the baroque’, claiming it is a style:

which deliberately exhausts (or tries to exhaust) all its possibilities and which borders on its own parody… [that] only too obviously exhibits or overdoes its own tricks.

He goes on to link this to a fundamentally comic worldview:

The baroque is intellectual, and Bernard Shaw has stated that all intellectual labour is essentially humorous.

I disagree. Having attended a big London exhibition about The Baroque I came away with the strong conviction that ‘the Baroque’ is above all about Power, the Complete Power wielded by monarchs who believed in their Divine Right to rule and the Total Power over all believers claimed by the Counter-Reformation Catholic Church.

Apart from anything else, Baroque works of art and churches are massive and imposing whereas Borges, if he is anything, is a precise miniaturist. He is more like a Swiss watchmaker than a Baroque architect.

But we are not reading Borges for accurate scholarship, in fact the precise opposite, we are reading him for his whimsical playing fast and loose with facts and figures and ideas for our amusement, an attitude he makes explicit when he writes that the stories are:

the irresponsible sport of a shy sort of man who could not bring himself to write short stories, and so amused himself by changing and distorting (sometimes without aesthetic justification) the stories of others.

Borges may be correct in using the term ‘Baroque’ to indicate an interest in following every detail or narrative possibility to its logical conclusion, in the compulsive inclusion of every finial and architectural flourish possible. But his work is at the opposite end of the scale from The Baroque style in art and architecture.

And the Baroque is, above all, deadly serious, whereas Borges’s work is informed throughout by a dry, metaphysical humour, that comes from somewhere else. This bookish humour is entirely Borges’s invention, filtered through the gentlemanly, ironic tone of the late Victorian British authors he loved so much.

Literary influence

Apparently (or, as Borges might write, ‘If I am not mistaken’) the Puerto Rican critic Angel Flores (1900 to 1994) was the first person to use the term ‘magical realism’ and dated the start of the Magical Realist movement from this book.

This is echoed by the blurb on the back of the Penguin edition which claims that Borges intended the stories simply to be light entertainments, newspaper squibs:

‘yet after its appearance in 1935 its influence on the fiction of Latin America was so profound that its publication date became a landmark in the history of Latin American literature.’

Borges reviews

A Universal History of Infamy (1935 trans. 1972)
Labyrinths (1962)
The Book of Imaginary Beings (1967, trans. 1969)
Dr. Brodie’s Report (1970)
The Book of Sand (1975 trans. 1977)
Jorge Luis Borges on Franz Kafka (1981)

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:

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.
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.
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.

Duodecimal Society of America

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

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.

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 a^bc^d, in the case of numbers with an odd number of digits the final one has no power, abcde’s powertrain is a^bc^de.

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’

Perfect, amicable and sociable numbers

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.

On-Line Encyclopedia of Integer Sequences

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 Golden Ratio

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 X_n+2= X_n+1 + X_n_.

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

The Fibonnaci sequence

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.

Hilbert’s Paradox of the Grand Hotel

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.

Reviews of other science books

Chemistry

The Periodic Kingdom: A Journey Into the Land of the Chemical Elements by Peter Atkins (1995)