Ulysses by James Joyce: introduction

‘You have the cursed jesuit strain in you, only it’s injected the wrong way.’
(Buck Mulligan arguing with Stephen Dedalus)

Hold to the now, the here, through which all future plunges to the past.
(Stephen’s credo)

‘Your own name is strange enough. I suppose it explains your fantastical humour.’
(John Eglinton responding to Stephen’s lecture about Shakespeare)

Fabulous artificer. The hawklike man…
(Stephen’s self-mocking self description in the same scene)

I’ll tickle his catastrophe, believe you me.
(Simon Dedalus threatening to write a letter to Buck Mulligan’s mother exposing him, and demonstrating his vivid and generally comic turn of phrase)

—I beg your parsnips, says Alf.
(In the Cyclops chapter)

James Joyce is a world class literary giant on the basis of his 1922 novel ‘Ulysses’. It’s monster long – around 700 pages in most editions – and has a fearsome reputation for being a ‘difficult’. In many senses it is difficult, often very difficult, but I’m going to have a go at explaining it as simply as I can.

Joyce’s previous and much more conventional novel, ‘A Portrait of the Artist as a Young Man’, (published in 1916) told the life story of a transparently fictional version of Joyce himself, named Stephen Dedalus, from toddlerdom to university. When described like that it sounds very straightforward; what complicates it is Joyce’s phenomenal intellectual powers and his increasingly experimental way with prose.

As in his set of short stories, Dubliners (published in 1912), on the surface all is realism, with realistic characters pottering round Dublin, getting into realistic scrapes and having realistic conversations. However, what was clear to perceptive readers of Dubliners from the get-go was that these stories are extremely carefully organised: at a meta level they are arranged so that the collection as a whole describes different stages of life – from boyhood japes in the early stories, to young manhood, maturity, through to the final story, titled The Dead. And within each story, there is also careful structuring and symbolism: for example, the short story ‘Grace’ opens with a middle-aged man passed out pissed in the toilets of a popular bar. He’s helped home by some mates and put to bed while his pals work with his wife to persuade him to go on a religious retreat to try and help him give up the booze. So far so mundane, until someone points out that the three locations of the story – downstairs toilet, bedroom and church – can be seen as the three locations of the afterlife: hell, purgatory and paradise. And once you know this, you are able to spot further little clues which have been sprinkled about the story, symbols or Latin phrases which subtly reference and gesture towards this concealed structure.

Well, magnify this method a thousandfold and you have ‘Ulysses’. I’ll consider it in three ways: first, the literal story; then the structure which underpins (or has been imposed) on it; thirdly, (some of) the linguistic innovations introduced in ‘Ulysses’, innovations which start slowly but spread to become completely rampant. It’s these innovations in prose style and structure which are the real stumbling blocks of the novel, often making it hard to read on the sentence, paragraph and page levels, presenting countless challenges to comprehension, and from relatively early on. But first, a look at the structure:

Structure

‘Ulysses’ is divided into 18 chapters, which are themselves gathered into three parts: part 1 contains 3 chapters; part 2 12 chapters; part 3 has 3 chapters.

What happens

Part 1: Chapters 1 to 3

At 8am on Thursday 4 June 1904 young Stephen Dedalus wakes up in the Martello Tower on Dublin Bay where he’s been dossing with a friend, medical student Malachi ‘Buck’ Mulligan. He’s cross because he was kept awake by the noisy nightmares of a third guest, the Englishman Haines. He’s also in a bad mood because Mulligan teases him because his (Stephen’s) mother recently died, and Stephen refused, on principle, to kneel by the bedside of his dying mother, something which now haunts him with guilt. This trio of young bucks go for a quick dip in the sea (well, fastidious Stephen doesn’t take part) then (in chapter 2) Stephen goes on to the school where he teaches part time, takes a history lesson, gets paid by the pedantic headmaster. In chapter 3 we are alone with Stephen and his thoughts as he walks along the beach.

Part 2: Chapter 4 to chapter 15

Cut back to 8am in the household of Leopold Bloom in central Dublin. Bloom is a middle-aged seller of newspaper advertising, a job which involves tramping the streets of Dublin touting for business. He’s married to Molly. He makes her breakfast in bed, fries breakfast for himself and then sets off on his day’s work. His day includes:

  • a trip to the Post Office
  • attending the funeral of a friend
  • visiting the office of a newspaper to place an ad
  • popping into Davy Byrne’s pub for a sandwich lunch
  • going to the National Library to look up an ad in an old newspaper, where his path doesn’t quite cross Stephen who is in the office of the Head Librarian, delivering another one of his literary theories, this time about the true meaning of Shakespeare’s Hamlet, to representatives of Dublin’s literary elite
  • Bloom has dinner at the Ormond Hotel while listening to other characters playing the piano and singing
  • Bloom moves on Barney Kiernan’s pub where he meets a character referred to only as The Citizen who is a pugnacious Irish nationalist who ends up trying to attack him
  • Bloom wanders out to Sandymount Strand on the seashore, and watches a young woman, Gerty MacDowell, who’s with some friends; aware of him watching her, Gerty deliberately flashes her legs in a sexy way and arouses Bloom so much that, watching from a safe distance as night falls on the beach, he masturbates to a climax: although it’s a little difficult to make out through Joyce’s highly-mannered prose what’s happening, it was this chapter which got ‘Ulysses’ banned in America and Britain
  • Bloom visits a maternity hospital where a family friend named Mina Purefoy is giving birth, and finally meets Stephen, who has been drinking with his medical student friends and is awaiting the arrival of his frenemy Buck Mulligan
  • when the maternity hospital drinking party breaks up, Stephen and his friend Lynch walk into Nighttown, Dublin’s red-light district, where Bloom follows them into Bella Cohen’s brothel; everyone is quite drunk by now and the scene is extremely long and filled with grotesque hallucinations, climaxing with Stephen being kicked out onto the street where he manages to get into an argument with a British soldier who knocks him to the ground where Bloom comes to his rescue

Part 3: Chapters 16 to 18

  • To sober him up, Bloom takes Stephen to a nearby cabman’s shelter by Butt Bridge where they encounter a drunken sailor
  • Bloom takes Stephen back to his place, makes him a cup of cocoa and they have a post-drunk conversation about the educational and cultural differences between them; they both go outside to pee in the garden; Stephen refuses Bloom’s offer of a bed for the night and staggers off into the night while Bloom goes to bed next to his sleeping wife
  • Chapter 18 is famous because it consists solely of Molly Bloom’s thoughts as she lies in bed next to her passed-out husband: the 40 or so pages contain only eight paragraphs with no punctuation in a tour-de-force of the relatively unknown technique of stream-of-consciousness: she remembers her various boyfriends and reminisces about courting and having sex with them, before the novel ends with a description of her having an orgasm, marked by the words yes yes yes which conclude this vast epic novel

There is One Big Fact I haven’t found space to explain yet and this is that, right from the start of his day, Bloom has known that a rival of his, the music impresario and flashy man-about-town, Hugh ‘Blazes’ Boylan, is going to call by his house, at 4pm that afternoon, supposedly to discuss details of the concert tour he’s arranged for Molly, but in reality to have sex with her. Bloom knows she is taking Boylan as her lover and yet feels powerless to stop it. And so he spends his entire day in a state of anxiety and suspense, continually looking at the clock at every venue he visits, in anticipation of zero hour; and then, after 4pm, reluctant to return to his house afterwards 1) lest he encounters the couple still at it of Boylan just leaving and 2) because he won’t know what to say to his wife.

So it’s a long book, and there’s a lot of words to read but I hope this summary shows that, on a basic narrative level, the story is relatively straightforward. All the events are highly realistic and plausible, if not actively boring and mundane, and once you’re told that this is the sequence of events the book describes, you can approach it with a lot less trepidation. What daunts people is the buried symbolism and above all the difficult prose style. Next: the Greek myth connection, or: why is it called Ulysses?

Ulysses and the Odyssey

Like every educated person, Joyce had read the two great epic poems of the ancient Greek author Homer, The Iliad and The Odyssey. The Iliad is a tragic account of the key episode at the heart of the ten-year-long Trojan War when the Greek hero Achilles, furious at the Greek leader Agamemnon, retires to his tent and refuses to fight. This has tragic consequences because when the Trojans counter-attack and make it as far as the Greek tents, Achilles lets his friend and soul-mate Patroclus put on his armour and rally the troops; Patroclus does this until he comes face to face with the Trojan hero Hector who slaughters him like a beast and the rest of the poem describes Achilles’ immense fury and bottomless grief.

But if the Iliad is tragic, the Odyssey has a very different feel. After the ten-year Trojan War ended, the Greek hero famous for his wily cunning, Odysseus, the man who came up with the idea of the Trojan Horse which led to the final defeat of the besieged city, it takes him ten further years to get home to his wife Penelope and his young son Telemachus (i.e. Odysseus is away from his kingdom of Ithaca for 20 years).

The poem actually opens near the end, with his last adventure, washing up on the shore of Princess Nausicaa, being discovered, bathed and dressed, then invited to a feast in her palace and it is here that he tells all the other guests his amazing adventures – being enslaved by Circe the magician, being held prisoner by the one-eyed Cyclops, sailing past the twin perils of Scylla and Charybdis, having to be tied to the mast in order not to give way to the seductive song of the Sirens, and many more.

Meanwhile, back in Ithaca, the Odyssey tells us how, throughout these ten long years, Odysseus’s faithful wife Penelope remained at home in their palace, putting off the many suitors who wanted to marry her and so inherit Odysseus’s kingdom. And it tells us how towards the end of this long period his son, Telemachus, come of age during his father’s absence, sets out on a quest of his own to find his father.

So with that understood, back to Joyce. Apparently Joyce had begun a story which he intended to be another short realistic yarn to join a revised edition of his short story collection ‘Dubliners’, about a Jewish advertising salesman with an unfaithful wife who wanders the streets in a peculiar frame of mind as he knows his wife is preparing to meet her lover, in his marital bed.

One thing led to another as Joyce wondered what if this advertising salesman met the young avatar he had created of himself in ‘A Portrait’, Stephen Dedalus? What would they discuss, what would they make of each other? And at some point he had the Eureka moment when it occurred to him that maybe this fellow Bloom’s wanderings around Dublin could be mapped onto Odysseus’s ancient adventures, maybe those ancient stories could give it a structure and, more than that, a kind of deep literary resonance, of the kind he was used to concealing in his Dubliners stories (like ‘Grace’, as described above),

When I tried to read ‘Ulysses’ at school I found it a great struggle because 1) I had no idea what was going on and 2) as the text progresses, the prose becomes difficult to read (see next section). You really need to know that the three central characters are modern avatars of Homer’s three characters: Bloom is wily old Odysseus; Stephen plays the role of his son, Telemachus; and Molly is the faithful Penelope waiting at home for her man. Except that they’re not a perfect match, are they? Stephen isn’t actually Bloom’s son and when, at the end, Bloom suggests a closer friendship Stephen mumbles something and wanders off into the night. No reconciliation there. And Molly, she is hardly the model of a faithful wife, in fact the whole point is that she is the exact opposite. And Bloom is hardly a macho Greek warrior, very much the reverse, he is a shy and diffident Jew in a Catholic country, liable to be discriminated against and picked on. So the novel’s central characters are in no way like-for-like matches of the Homeric epic, they are something more like ironic reincarnations, satirical avatars, reflecting the comic bathos of modern life.

But if I didn’t understand the roles played by these characters, the biggest single stumbling block to grasping the mythic resonances of the novel was the way the 18 chapters in Joyce’s text have no titles. Just giving them titles, and a sentence of explanation, would have transformed my experience.

As soon as I came across a book which explained that each of the chapters reflects or is based on a specific episode from the Odyssey, and clearly indicated which one was which, my whole reading experience was transformed. Augmented by the knowledge that the 18 chapters are further grouped into three parts which themselves are based on the main thematic elements of the Odyssey story, as follows:

  1. the first three chapters, describing Stephen waking in the tower, going to school and wandering into Dublin, are titled the Telemachia or wanderings of Telemachus
  2. the 12 central chapters can be thought of as the Odyssey proper, describing the many adventures of Bloom during the day
  3. the final three chapters, when Bloom takes Stephen home, can be grouped as the Nostos, Greek for return so the ‘Return’ part of the story, reflecting the final chapters of Homer’s poem which describe the homecoming of the long-absent hero

Why Ulysses and not Odysseus?

The ancient Romans not only co-opted many of the Greek gods, subsuming them into the existing Latin pantheon, they did the same with many of the mortal heroes of ancient Greece. When the Romans translated the Greek legends from Greek into Latin, they sometimes found it easier to change the names as well, to make them easier to pronounce. The early Latin translators of Homer probably adapted the Greek Ὀδυσσεύς into the Sicilian Οὐλίξης, and then the Etruscan Uluxe, before arriving at the shorter, more Roman-friendly Ulysses.

So that’s why you see two names being used for the same person: Odysseus is his original Greek name as used by Homer; Ulysses is the name used by Roman authors, such as Virgil, when describing the Tale of Troy.

So why did Joyce use the Latin name? Apparently, when he was a boy of 12, Joyce’s thoughts on the subject were crystallised when he first read the story in Charles Lamb’s book ‘The Story of Ulysses’ (1808), and that version of the name stuck. Having been brought up in a heavily Catholic school, Latin was all around him, in the Mass and liturgy and so on, and so he took to the Latinate forms (unlike late Victorian English Protestant private schools of the time, where there would have been as much emphasis on ancient Greek and so the Greek names are preferred by English writers).

Plot structure version 2

So here’s the chapter structure again, but this time indicating the episode from the Odyssey which each one is based on and explaining the parallels with the Homeric episode, such as they are:

Part 1: The Telemachia (the wanderings of Telemachus / Stephen)

Chapter 1: Telemachus

8am. Introducing Stephen Dedalus, bunking in the Martello Tower being rented by Buck Mulligan. Breakfast. Stephen should be mourning his recently dead mother and so borrows an all-black suit from Mulligan. In other words, he is dressed in mourning for the whole of the novel.

Chapter 2: Nestor (wise king of Pylos, advisor to the heroes)

Stephen’s encounter with school headmaster Garrett Deasy, who asks him to take a letter to the newspaper about foot-and-mouth disease. (Which gives rise to Stephen’s mortified expectation that Mulligan will mock him as ‘the bullockbefriending bard’, which becomes one of many recurring phrases, not to mention later cattle-related jokes such as someone being a bull in a china shop etc.)

[Myth parallel: Remember I mentioned that Bloom, Stephen and Molly are not direct avatars of their heroic predecessors but more like satirical, ironic modern versions. Well, as you can see here, the pompous Unionist headmaster Deasy is far indeed from being an avatar of wise old king Nestor of Greek mythology. Quite obviously he is a comic parody and yet there are commonalities: Nestor trained young horses, Deasy’s career is training teenage boys.]

Chapter 3: Proteus (the shape-shifting sea god from Greek myth)

Stephen wanders down to the seashore at Sandymount Strand and his mind wanders, free-associating ideas and memories from school and university, shreds of knowledge, quotes in various languages, perceptions and the rhythmic sound of the waves all melding in his mind to make this one of the most daunting of all the chapters to read, certainly in the first half.

[Myth parallel: You can see how the concept of the shape-shifting god maps nicely onto the endless shape-shifting of the protagonist’s mind.]

Part 2: the Odyssey (the wanderings of Ulysses / Leopold Bloom)

Chapter 4: Calypso

Cut back to 8am and Leopold Bloom, waking up, having breakfast, reading his mail, taking a letter from Blazes Boylan to his wife Molly in bed.

[Myth parallel: Calypso was a nymph from Greek mythology, famous for holding the hero Odysseus captive on her island, Ogygia, for seven years. Note that here, at the start of Bloom’s novel, Molly is Calypso. In the final chapter the same Molly stands in for the completely different figure of Penelope. Which shows you how Joyce’s parallels are exact in some respects but can be very loose if needs be.]

Chapter 5: Lotus Eaters

Bloom has an hour to kill before he has to attend Paddy Dignam’s funeral at 11am so he does a handful of chores in a lazy sensual mood: he goes to the Post Office, posts a letter, wanders into a Catholic church (incense and gold), buys a bar of lemon soap at a chemist and fantasises about paying a visit to Dublin’s Turkish baths.

[Myth parallel: The Lotus Eaters were a mythical people from Homer’s Odyssey, living on an island where they ate the narcotic lotus flower which caused blissful forgetfulness, a desire to stay, and loss of ambition. This is one of the more oblique of the Homeric parallels.]

Chapter 6: Hades or hell

Along with three friends, including Stephen’s father Simon Dedalus, Bloom attends the funeral of Paddy Dignam, taking the long journey by funeral carriage across Dublin to the burial in Glasnevin cemetery. This triggers in Bloom thoughts of his own son, Rudy, who died young, and of his father who committed suicide with poison.

[Myth parallel: This is one of the more obvious and direct allusions, matching Odysseus’s trip to the Greek underworld and a modern funeral.]

Chapter 7: Aeolus, the god of wind

This chapter is notable for the way the text is broken up by no fewer than 63 captions in the style of newspaper headlines, ironically summarising the content of each section.

At the office of the Dublin newspaper, the Freeman’s Journal, Bloom walks past the huge printing machines to the editor’s office where he attempts to place an ad for a client. Stephen arrives at the same office bringing (as promised) Deasy’s letter about foot-and-mouth disease. Bloom notices Stephen and that his boots are dirty but they do not actually meet. Instead the chapter mostly consists of long wordy conversations between half a dozen editors, hacks and scroungers, about each other, Dublin gossip, and racing tips.

[Myth parallel: In Greek mythology Aeolus was the keeper of the winds, ruling from the island of Aeolia where he held the violent winds captive in a bag. Control of the winds of information and opinion is an ironic or satirical way of thinking about a newspaper, and gassy windiness also describes the banter of the 7 or 8 characters gathered in the office.]

Chapter 8: Lestrygonians (a race of giant, cannibalistic ogres in Greek mythology)

Bloom wanders the streets feeling hungry, bumps into an old flame, Josie Breen, notices the (real life) author A.E. walking past him with a lady acolyte. He enters Burton’s restaurant but is overwhelmed by the smell and sight of so many people eating, so leaves and pops into Davy Byrne’s pub for a light lunch of a cheese sandwich and a glass of Burgundy wine. When he leaves, 4 or 5 other characters discuss his character.

Out on the street, he helps a blind man (who we later learn is a piano tuner) to cross the road and ponders at length all the implications of being blind – then walks on to the National Library. Thinking about food has led him to consider the human body as a machine, food in-poo out, which leads to the eccentric speculation whether traditional Greek statues were depicted with anuses. As he enters the National Library he has a quick squint up at the big statues flanking the entrance, to check.

[Myth parallel: obsession with food links the classical reference and the modern chapter.]

Chapter 9: Scylla and Charybdis

Cut to Stephen in the head librarian’s office at the National Library delivering another one of his literary theories, this time about the true meaning of Shakespeare’s ‘Hamlet’, appropriately enough because he is dressed in mourning (though for a dead mother not a dead father, as is the case with the Danish prince). Stephen and Bloom’s paths nearly cross as the latter pops in to look up an ad in an old newspaper but, again, they don’t actually meet.

[Myth parallel: In the Odyssey Odysseus’ ship has to sail the narrow channel between the twin monsters Scylla and Charybdis. I’ve read a clever interpretation pointing out that the two monsters can be mapped on to the two types of aesthetic theory dramatised in this chapter: Stephen’s theory of Shakespeare insists that the Bard’s great plays arose from the tribulations of his own sometimes squalid personal life but he’s presenting it to the older generation of Dublin critics who believe art should be about beautiful timeless spiritual ideals – so it’s these two opposing theories which represent the two legendary monsters, and which Joyce the author had to navigate between. Clever. There might also be a canny little micro-parallel because, as Bloom enters the Library, he walks between Stephen and his friend Mulligan exiting, who also represent two ends of a spectrum, Mulligan all glib flashy smartness, Stephen, deep but gloomy introspection.]

Chapter 10: Wandering Rocks

Nineteen short vignettes describe the movements of the central figures and about 30 secondary characters through the streets of Dublin. The episode begins by following kind-hearted Father Conmee, a Jesuit priest, walking north and stopping for a kind word for various parishioners, and ends with an account of the cavalcade of the Lord Lieutenant of Ireland, William Ward, Earl of Dudley, processing through the streets on his way to inaugurate the Mirus bazaar in aid of funds for Mercer’s hospital, and with the response of the 19 or so characters described in the preceding vignettes to his carriage as it passes.

[Myth parallel: Wandering rocks and wandering Dubliners.]

Chapter 11: Sirens

Bloom has dinner at the Ormond Hotel whose bar is dominated by the ministrations of two barmaids, dark-haired Miss Kennedy and brunette Miss Douce, while other characters gossip and then play the piano and sing some airs in the hotel’s dining room. Bloom gets a pen and paper to write a furtive reply to his lover Martha’s letter to him.

[Myth parallel: In Greek mythology the sirens were enchanting beings, half-woman, half-bird, who lured sailors to shipwreck and death with their irresistible songs; so the sirens are represented by the two barmaids, who don’t themselves sing but the scene is full of characters playing and singing, not least Stephen’s father, Simon.]

Chapter 12: Cyclops

Up till now the chapters have been cast in what you could call basic stream of consciousness, albeit often difficult to follow. The remaining six chapters of the novel are all longer and each one has its own individual format. Chapter 12 breaks the convention of the novel up to now by being narrated by an unnamed Dubliner who has a completely different ‘voice’ or style from either Stephen or Bloom.

This narrator works as a debt collector and goes to Barney Kiernan’s pub where he meets a character referred to only as ‘The Citizen’ who is a vehement Irish nationalist. When Bloom arrives at the pub, he is subjected to an antisemitic rant by the Citizen and various other characters come to his defence.

[Myth parallel: The Cyclops in Homer’s Odyssey was Polyphemus, a giant, one-eyed son of the sea god Poseidon who captured Odysseus and his men in his cave. This monster eats several of the sailors before Odysseus manages to get him drunk then blind him with a sharpened stake, and escaping. So the Citizen is the Cyclops and the one-eyed reference is a satirical take on the bigoted monomania of Irish nationalism.]

Chapter 13: Nausicaa

Bloom wanders out to Sandymount Strand where he watches a young woman, Gerty MacDowell, who flashes her legs to excite him. The first half is written as a parody of a romance magazine, the second half more realistically from Bloom’s point of view.

[In Homer’s Odyssey, Nausicaa was daughter of King Alcinous and Queen Arete of Phaeacia, famous for discovering and helping the shipwrecked Odysseus, giving him clothes, food, and guidance to her father’s palace. So the reincarnation of the elegant princess as a flirty young woman is obviously full of ironies and a vivid example of the general idea of how ‘fallen’ or degraded the modern world is.]

Chapter 14: Oxen of the Sun

Bloom visits the maternity hospital where Mina Purefoy is giving birth, and finally meets Stephen, who has been drinking with his medical student friends. At the end the drinking party breaks up, Mulligan catching the train back to his tower while Stephen drunkenly blunders off to the red light district accompanied by his pal, medical student Vincent Lynch. Worried about him, a much more sober Bloom follows.

[In the Odyssey Odysseus and his men come to the island of the sun god Helios, and are warned not to touch his holy cattle. When Odysseus goes off to pray for guidance his hungry men slaughter, roast and eat some of the cattle. Odysseus returns and tells them to flee but their ships are pursued by the sun god’s anger and most of his men are wrecked and drowned. The parallel is loose here, as this chapter is famous for mimicking the growth of the foetus in its mother’s womb by parodying the evolution of English from the original Anglo-Saxon onwards.]

Chapter 15: Circe

Bloom follows Stephen and his pal Lynch into a brothel where they both experience a series of grotesque Rabelaisian hallucinations. After smashing the chandelier with his walking stick, Stephen is kicked out onto the street where he gets into a fight with a British soldier and is knocked down, leaving Bloom to pick him up and sort him out.

[In the Odyssey, Circe was a powerful sorceress who used her magic to turn Odysseus’s men into swine and keep them imprisoned for years. Obviously there are no years here, just one night, and there is no crew, just Stephen, Lynch and Bloom. But the monstrous brothelkeeper Bella Cohen is a fittingly grotesque parody of the beautiful magical Circe of Homer.]

Part 3: Nostos (the Return)

Chapter 16: Eumaeus

[Myth parallel: in the Odyssey, Eumaeus is the loyal swineherd of Odysseus, a man of noble birth kidnapped as a child who had become a trusted friend of the master and is the first point of contact for the disguised hero upon his return to Ithaca.]

In the novel, having saved Stephen from arrest after his altercation with the soldier, Bloom takes Stephen to the cabman’s shelter near Butt Bridge where they encounter a drunken sailor: so the drunken sailor is the parallel to Eumaeus.

Chapter 17: Ithaca

[Myth parallel: In the Odyssey, Ithaca is Odysseus’s kingdom to which he returns.]

In the novel Ithaca it is Bloom’s house, to which he brings Stephen for a cup of cocoa.

Chapter 18: Penelope

[Myth parallel: In the Odyssey, Penelope is Odysseus’s loyal wife who waits for 20 long years for her husband to return, spurning the advances of numerous eligible suitors to replace the husband they all claim is dead.]

In the novel, Penelope is reincarnated as Molly Bloom, and the novel ends by abandoning the men and the male perspective altogether and ending with the huge seamless stream-of-consciousness flow of Molly’s falling-asleep thoughts.

Technical innovations

So far I’ve talked about the ‘structure’ of the novel but I have barely mentioned the technical or style innovations which make it such a demanding read on the page, made it so notorious in its time, and for a generation afterwards made it feel as if Joyce had not just revolutionised the novel with his elaborate system of symbols and references, but had revolutionised English prose as well.

Two or three of these quirks or innovations had already appeared in ‘A Portrait’ but in ‘Ulysses’ they are cranked up to the max, along with new novelties.

1. Formal, studied prose

Not an innovation as such, just the foundation on which everything else sits: but Joyce’s prose, even at its least experimental, is surprisingly formal and stiff and mannered. He consistently writes sentences not as you’d expect them to flow but with a deliberate stiffness. He consistently puts words into a counter-intuitive order. After a while I realised he always puts the adverb where you don’t expect it, counter to its usual position in everyday English. As in: ‘He waited by the counter, inhaling slowly the keen reek of drugs…’ where putting ‘slowly’ after ‘inhaling’ makes you linger on it longer, process it more, gives it more weight. Or:

The carriage turned again its stiff wheels and their trunks swayed gently. Martin Cunningham twirled more quickly the peak of his beard.

You’d expect ‘again’ to come after ‘stiff wheels’; you’d expect ‘more quickly’ to come after ‘beard’ and so on. He does this even in his plainest sentences and it gives them a studied, calculated movement. You argue he does so in order to describe things, especially people’s actions, with a finicky super-precision. Here’s the start of chapter 6:

Martin Cunningham, first, poked his silkhatted head into the creaking carriage and, entering deftly, seated himself. Mr Power stepped in after him, curving his height with care.

Also, he enjoyed avoiding the common word and using the slightly more official or officious word or phrase. After helping him with his sums, Stephen watches Sargent hurriedly change and run out onto the sports field.

He stood in the porch and watched the laggard hurry towards the scrappy field where sharp voices were in strife. They were sorted in teams and Mr Deasy came away stepping over wisps of grass with gaitered feet. When he had reached the schoolhouse voices again contending called to him.

‘Where sharp voices were in strife‘ is not the easiest most colloquial way to describe a bunch of boys shouting on a football pitch. It is studied and formal. Ditto the third sentence which contains the odd phrase ‘voices again contending’. You or I might write ‘the sound of the boys arguing again’ but Joyce prefers this much more stiff and formal arrangement: ‘voices again contending’. And note ‘again’ in the unusual position, you or I would say ‘voices arguing again’, but putting it before the verb, this time, has the effect of making every word feel more studied and carefully presented, as at an exhibition of sentences.

So we must bear in mind that even when he’s trying to write relatively ‘straight’, before he got up to any formal tricks, Joyce’s prose style was already oddly stiff, spavined and constricted: highly self-conscious and ornately arranged. This lends even the most supposedly straightforward passages a certain stiff, presentational feel, before we get to any of his party tricks.

2. No speech marks

Joyce had a foible about/well thought-out intellectual objections (delete where applicable) to speech marks / quotation marks / inverted commas. In all the texts Joyce had final say over he replaced the conventional introduction of speech by double apostrophes with an em dash, with no indication where a piece of dialogue ended. Just this one change is surprisingly confusing. It has the cumulative effect of meaning you’re never quite sure where a piece of speech ends and the narrative, or a character’s thoughts, begin.

—Three, Mr Deasy said, turning his little savingsbox about in his hand. These are handy things to have. See. This is for sovereigns. This is for shillings. Sixpences, halfcrowns. And here crowns. See.

3. No hyphens

Just as he disliked speech marks, so Joyce early on decided to dispense with hyphens and just to run two hyphenated words together. ‘A Portrait’ is full of examples like illfated, selfrestraint, rosesoft and hundreds more. And so it is here, as indicated by the novel’s famous opening sentences:

Stately, plump Buck Mulligan came from the stairhead, bearing a bowl of lather on which a mirror and a razor lay crossed. A yellow dressinggown, ungirdled, was sustained gently behind him on the mild morning air.

Innocent though this foible first appears, like dispensing with speech marks it is the first stirrings of the disintegration of language the book is going to deploy on a massive scale. For just as dispensing with speech marks makes it increasingly hard to know where direct speech ends and free indirect speech (i.e. the character’s own thoughts) begins; so dispensing with hyphens where they ought to go marks the start of start of not knowing where one word ends and another begins; in practice, it marks the start of Joyce’s running words together in original and increasingly inventive ways.

It starts with dropping hyphens in a phrase like:

He passed an arm through the armstrap and looked seriously from the open carriagewindow at the lowered blinds of the avenue.'(chapter 6).

But then it moves on to sticking together words which should never be joined to create new words:

… an old woman peeping. Nose whiteflattened against the pane

You can see in this example how he realises he’s stumbled across a new piece of grammar, the portmanteau noun, and as the novel progresses, the technique of jamming 2, 3 or more words together becomes more outrageous. So as Stephen walks by the sea in chapter 3, he fancifully imagines everyone in the world linked back to their mothers via a ghostly umbilical cord:

The cords of all link back, strandentwining cable of all flesh…

And then drops the first of his made-up, portmanteau words, imagining the early Christian heretic Arius:

Warring his life long upon the contransmagnificandjewbangtantiality.

After all, if you can glue two words directly together, why not three or four or five words? This is a taste of things to come…

3. Learnèd allusions

We could be here all week describing this one but the basic idea is simple. Joyce was hyper well-read and developed the habit in ‘A Portrait’ of dropping allusions to his learning into the narrative bits of text. Quite regularly the supposed narrative in fact contains no narrative at all, just a tissue of allusions, as if giving you direct access to the flux of (super-literate) thoughts in the main protagonist, Stephen’s, head. So you have potentially three elements: 1) old-fashioned third-person narration; 2) speech without speech marks; 3) the protagonist’s thoughts reflected in indirect speech.

Example 1

The opening of chapter 2 demonstrates all three elements: both the abolition of speech marks, a brief appearance from a conventional narrator, then an abrupt jump into Stephen’s hyper-educated mind.

(1: no-speech-marks speech)
—You, Cochrane, what city sent for him?
—Tarentum, sir.
—Very good. Well?
—There was a battle, sir.
—Very good. Where?
(2: third-person narrative)
The boy’s blank face asked the blank window.
(3: straight into Stephen’s thoughts)
Fabled by the daughters of memory. And yet it was in some way if not as memory fabled it. A phrase, then, of impatience, thud of Blake’s wings of excess. I hear the ruin of all space, shattered glass and toppling masonry, and time one livid final flame. What’s left us then?

There has been no narrative lead-in or introduction, no text explaining that we are in a school classroom and Stephen is taking a lesson. Only from the dialogue can we deduce this is what is happening and that it is a lesson about ancient history. ‘The boy’s blank face asked the blank window’ is a neat way of describing the poor schoolboy’s blank ignorance of Stephen’s question, but look what happens next.

We are thrown straight into Stephen’s ‘stream of consciousness’. I admit I had to stop and puzzle this out and have only got parts of it. The two sentences using ‘fabled’ I think reflect Stephen thinking that the battle referred to has gone down in history and yet, he reflects, was probably not as ‘memory’ tells fables about it.

What battle? He will go on to quote the Roman General Pyrrhus who won a battle in 279 BC where the Romans losses were so bad he is supposed to have said: ‘Another victory like that and we are done for.’ This is where we get our phrase ‘A Pyrrhic victory’ from. Stephen is (I think) reflecting that this phrase expressed not so much the general’s despair as his ‘impatience’.

I know the poet William Blake wrote in praise of excess: in his ‘Proverbs of Hell’ he wrote that ‘The road of excess leads to the palace of wisdom.’ Maybe Stephen is conflating the disastrous outcome of the Roman battle, its excessiveness, with the ‘wisdom’ contained in the general’s phrase which led to it becoming a proverb.

As to the next bit, ‘I hear the ruin of all space, shattered glass and toppling masonry, and time one livid final flame’ I don’t think this is particularly tied to Pyrrhus or this history lesson, although I know it is a phrase which will be repeated throughout the novel at various moments. So it is more like an early appearance of an idée fixe of Stephen’s and, if we consider ‘Ulysses’ as a work of art, it is like a motif which is being introduced early on so that it can be repeated with variations later on.

There’s probably more going on here than I’ve indicated and I might have got some of this wrong, but you see how it works: just elements – bits of dialogue which are not at all clearly demarcated, minimal amount of narrative explanation, then chunks of Stephen’s internal monologue which is ferociously learned and allusive – are already combining to make it a tricky read.

Don’t panic

As a bookish person, who’s read a lot of the same books as Joyce, I get some of his references and/or I’ve taken the trouble to look (some of) them up – but there is one key principle to bear in mind here, which is: Don’t be afraid.

Tens of thousands of academics have spent their entire lives elucidating ‘Ulysses’ and nobody has got all the allusions buried in it. It doesn’t matter. If you like puzzles, you can stop at each paragraph and look up the allusions. Or you can read the novel with a page-by-page guide (online or hard copy) open beside it to explain them. (If you have the patience, that’s probably the way to get the most out of reading ‘Ulysses’.)

As Canadian academic Hugh Kenner puts it, the book’s innumerable correspondences and patterns ‘adds fun to our endless exploration of this book’ – if, that is, endless exploration of a vast tissue of learnèd references and internal echoes is your idea of fun.

But if you’re not that kind of person, don’t worry. Read at the book, forge on through it, and let its unusual methods creep up on you. At various points you’ll recognise the same quotes or allusions cropping up and begin to get a feel for them, how they recur and give structure to the text, like motifs returning in a long piece of music. As in the ‘Cantos’ of Ezra Pound you don’t even have to understand what they mean (quotes from foreign languages, for example) for their repetition to start to have a haunting and evocative effect.

Also: it is as well to be clear that Stephen is not a god, he is not the prophet of some challenging religion: he’s just a character in a book, and his character is that of a cleverclogs, a callow young man too clever by half. He’s read all the books in the world but has little or no life experience, and it shows. Therefore, to some extent his thoughts are probably intended to be offputtingly clever-clever.

Example 2

Here’s another example of the method. Stephen stays after class to help a poor schoolboy, Sargent, with his sums. But as he does so, his overworking, over-educated intellect reflects that the symbols used in algebra are Arab in origin, in fact the word ‘algebra’ is itself Arabic, and this prompts him to think of the two great medieval philosophers – Ibn Rushd (1126 to 1198) Latinized as Averroes, and Moses ben Maimon (died 1204), commonly known as Maimonides. This is at least part of what is going on in this passage:

Across the page the symbols moved in grave morrice, in the mummery of their letters, wearing quaint caps of squares and cubes. Give hands, traverse, bow to partner: so: imps of fancy of the Moors. Gone too from the world, Averroes and Moses Maimonides, dark men in mien and movement, flashing in their mocking mirrors the obscure soul of the world, a darkness shining in brightness which brightness could not comprehend.

The dance of symbols across the page is described as a ‘morrice’ which is an antique adjective for Moorish or Arab, but has echoes of Morris dancing, a connotation reinforced by the word mummery, which is an olde English word for acting. Both of them clearly refer to the algebraic symbols Stephen is trying to teach the boy Sargent about and which he fancifully envisions dancing hand in hand with each other, bowing to their partners and so on.

The second half of the paragraph is, as far as I can see, a poetic evocation of the effect of the medieval scholars’ writings, which was itself so complicated and full of learned allusions that Stephen envisions it as mirrors. Maybe the two wise men’s learning is referred to as a darkness because 1) they were both dark-skinned non-European men and 2) maybe Western Europe is the brightness and lightness in which their complex, dark-skinned wisdom made little impression. I’m not sure. Something like that…

Anyway, this kind of thing happens thousands and thousands of times throughout the book. It makes up most of the long the novel’s texture, so it helps if you yourself are bookish and like spotting allusions. But, as I keep emphasising, it’s not absolutely necessary to get every allusion to enjoy the book, in fact it’s probably impossible. It’s perfectly valid to read the whole thing without ‘getting’ any of the allusions because there is plenty of other stuff going on – the structure of the plot itself (as outlined above) but also tens of thousands of places where the prose is so unexpected and inventive that you can enjoy it on the surface, for it mysteriousness and multitudinous rhythms, as much as for this riddling Sudoku element.

The internet / AI changes everything

As I wrote the preceding paragraphs it began to dawn on me that nowadays, of course, the whole experience of reading, especially reading difficult or demanding books, has been transformed by the internet and not just the old internet but the shiny new world of artificial intelligence.

Nowadays if you’re puzzled by anything in ‘Ulysses’, from the overall structure to the tiniest word, you can ask an LLM like Chat GPT and chances are it will explain everything. For example, I was not understanding the scattered references to Bloom’s father in chapter 6 (Hades) and so I simply asked Chat GPT: ‘In James Joyce’s Ulysses, did Leopold Bloom’s father commit suicide?’ and Chat came right back with ‘Yes’ along with details such as the method (poison) and location (the Queen’s Hotel).

So there’s now the facility to look up everything – from the granular level of individual words, foreign quotes, odd phrases, through to the macro level of my Bloom query – on an AI (Chat, CoPilot, Gemini) and have answers delivered on a plate.

Whether this is an appropriate way to read the book, whether it short circuits the time and effort Joyce intended his readers to invest in it, whether it undermines the experience of slowly constructing your own version from the fragments you notice or understand, and replaces it with a fully explained, Sam Altman-friendly version, is open to debate. But there’s no denying AI’s help in immediately solving thousands of niggling details or impenetrable obscurities, for example: who is Hynes? What does ‘De mortuis nil nisi prius’ mean? Can I find a recording of ‘Those lovely seaside girls? (Yes – https://www.youtube.com/watch?v=O4IpDMyox2Y).

After a while I began to rely on it more heavily because the broken-up prose and elliptical style of so many of the conversations often mean it’s very hard to figure out what’s going on, and even what people are saying to each other.

But academic books will remain useful…

Where books, even quite old guides to ‘Ulysses’, score over the internet, is that they will offer useful and interesting opinions and insights. Chat will 1) only answer the question you asked; it might answer it fully and give you more detail than you expected, but at the end of the day the answer is limited by how you phrase your initial question or ‘prompt’. And 2) it will only give you other people’s opinions, neatly summarised and tied up in a bow.

By contrast a book-length guide will tend to introduce you to ideas and interpretations you’d never thought of before. They let you share, and follow the logic behind, distinct and maybe idiosyncratic interpretations, by expert scholars. LLMs tend to repeat and confirm the biases or expectations you bring with you whereas (good) books open the mind to all kinds of new possibilities.

So AI has already revolutionised the process of reading difficult works of literature like ‘Ulysses’ (and many more). But good books of criticism or analysis or just good quality guides, will for the foreseeable future still have the advantage of opening your mind to new ideas. Which, back in the olden days, was often considered an element point of studying literature…


Credit

‘Ulysses’ by James Joyce was published by Shakespeare and Company in 1922.

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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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