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The Romantic by Hermann Broch (1931)

It was only fragments of the past that fleetingly emerged, and important and trivial things flowed chaotically through one another… (The Romantic, page 11)

Hermann Broch (1886 – 1951) is considered one of the major European Modernist authors. He was born in Vienna to a prosperous Jewish family and worked for some time in his family’s factory. In 1909 he converted to Roman Catholicism and married Franziska von Rothermann, the daughter of a knighted manufacturer. In 1927 i.e. aged 40, Broch sold the textile factory and decided to study mathematics, philosophy and psychology at the University of Vienna, and to pursue a full-time career as a writer. At the age of 45, in 1931, his first major literary work, the trilogy The Sleepwalkers, was published in Munich.

The Sleepwalkers consists of three medium-sized novels:

The Romantic (1888)
The Anarchist (1903)
The Realist (1918)

The dates are not my addition, they’re part of the formal, full titles of each novel, indicating:

That each novel is, among other things, a portrait of its era
That Broch is quite a schematic writer. Recall that he chose to study maths at university. Note that 1888 to 1903 is 15 years, and 1903 to 1918 is 15 years. So a span of 30 years. And it is symmetrical. And it is a trilogy, suggesting three points of focus…

Reading Hermann Broch

I read the trilogy in the English translation made by prolific translators Willa and Edwin Muir soon after the original German publication, back in the early 1930s.

There’s no getting round the fact that Broch is pretty difficult to read, for a number of reasons:

Long paragraphs Weaned on a hundred years of post-Hemingway minimalism, Anglo-Saxon readers are used to short sentences in short paragraphs. Whereas Broch – like Kafka – routinely deploys paragraphs which last an entire page, sometimes two, sometimes even more, so that the reader is confronted by what initially appears to be a wall of words.

In the modern Anglo-Saxon tradition, dialogue is broken up so that each exchange starts on a new line, making it visually and psychologically easy to follow. Not here. Extended dialogues are presented as unbroken blocks of text, which can make them hard to follow. If your focus drifts at all, it’s quite easy to find you’ve ‘read’ an entire page with absolutely no memory of what happened.

Long sentences The very long paragraphs contain some very, very long sentences. Routinely I got into the habit of having to reread entire paragraphs, and certainly some of the half-page-long sentences. Rereading helped them swim up into meaning.

The translation In almost every sentence there are ungainly and sometimes grammatically questionable turns of phrase.

Besides, visiting Berlin but twice a year, he had abundance to do when he was there. (p.11)

Perhaps his mother was really against his being sent to Culm, but one could put no dependence on her. (p.13)

Nevertheless she resolved to ask Joachim some time what was his birthday. (p.74)

Is this because German has such a different language from English, and the Muirs have stuck as close as possible to German word order? Or is it because Broch’s ‘Modernist’ German would be difficult even for a German speaker and the translators have tried to capture that difficulty?

There is no real way of knowing, but reading Broch is emphatically not like reading an English author.

Difficult descriptions Some of the text swims into view and suddenly you understand what is going on, who is talking, and what they’re saying. Then at other moments the text becomes blurry, describes the characters’ confused emotions or intuitions or misperceptions even, at moments (particularly when seen through the eyes of the central character, Joachim von Pasenow) what seem almost to be hallucinations.

Yet now suddenly everything had receded to a great distance in which Ruzena’s face and Bertrand’s could scarcely be told from each other. (p.56)

A lot of the time you’re not sure whether this is carefully calculated effect, or the cumulative impact of the long sentences in long paragraphs rendered into unidiomatic English. Is it you or him?

Stream of consciousness After a while I began to realise that, at least in part, it’s him i.e. it is a calculated effect. As you get used to Broch’s ‘background’ style, you begin to be able to make out passages where the characters have giddy, dizzy moments of misperception, the central character, Joachim von Pasenow, in particular being subject to all kinds of odd and confusing thoughts.

Things were as elusive as a melody that one thinks one cannot forget and yet loses the thread of, only to be compelled to seek it again and again in anguish. (p.114)

And you realise that at least part of Broch’s intention is to capture the flow of thoughts, and the evanescence of consciousness. Broch takes us into the mind of Joachim, and then of the two other central characters, in order to show us how multi-levelled consciousness is, and how often half-formed ideas or impressions float across our minds without ever coming into focus, often because we don’t want to fully acknowledge them.

Phenomenology I wonder what kind of philosophy Broch studied at the University of Vienna because this focus on trying to describe the actual processes of consciousness – the flavour of different thoughts, and the ways different types of thought arise and pass and sink in our minds – reminds me that Phenomenology was a Germanic school of philosophy from the early part of the century, initially associated with Vienna.

In its most basic form, phenomenology attempts to create conditions for the objective study of topics usually regarded as subjective: consciousness and the content of conscious experiences such as judgements, perceptions, and emotions. Although phenomenology seeks to be scientific, it does not attempt to study consciousness from the perspective of clinical psychology or neurology. Instead, it seeks through systematic reflection to determine the essential properties and structures of experience. (Wikipedia)

That’s not a bad summary of what Broch appears to be doing in this novel. Here’s how one character feels about ‘love’:

It was an almost joyful ground for reassurance that the feeling which she hopefully designated as love should have such a very unassuming and civilised appearance; one had actually to search one’s mind to discern it, for it was so faint and thin that only against a background of silvery ennui did it become visible. (p.70)

Just one example of from hundreds of a Broch character seeking, searching to define and make out feelings or ideas or notions which hover on the edge of consciousness or definition.

Novel of ideas The ‘novel of ideas’ is a notoriously slippery concept to define. This is more of a novel with ideas.

This is most obvious in the cleverest character, Eduard von Bertrand, who makes subtle, sophisticated or ironic speeches about love or religion or the notable speech about African Christians over-running Europe (see below).

But other characters also struggle to define and understand ideas. When Elisabeth is at home with her parents in the country there is a two- or three-page passage where she reflects at length on the special nature of her parents’ marriage, which includes a meditation on the true meaning of collecting, of making collections (to overcome death, p.71) and of age, which she experiences not as an idea but like a serpent stifling her consciousness.

Joachim spends his entire time trying to sort out his ideas about honour, duty, the army, the uniform and so on and, in the final third or so of the novel becomes obsessed with religious imagery, with a conviction of his own sinfulness and that God is punishing him (what for? well, that’s what he spends his time agonising about).

But the most philosophical character is the narrator. Personally I feel the novel gets off to a rocky start, we are introduced to too many characters in a quirky and almost incomprehensible way. But once it beds in, you are never more than a few pages from an extended description which tends to morph into ‘philosophical’ thoughts about many aspects of the quiet bourgeois style of life the book describes: the effect of the music Elisabeth plays in a piano trio (p.92) or an extended description of the landscaping of the garden round the Baddensens’ manor house, or so on.

It is not a novel of ideas in the sense that it proposes a massive concept of society like 1984 or is full of clever character sitting round discussing Important Subjects, as in an Aldous Huxley novel. It is more a novel which describes the complex feelings and intuitions of its characters which sometimes invoke larger ideas or notions. In one scene Elisabeth and her mother pay a visit to the von Pasenows. The conversation is getting a little rocky when the pet canary starts singing.

They gathered round it as round a fountain and for a few moments forgot everything else; it was as though this slender golden thread of sound, rising and falling, were winding itself around them and linking them in that unity on which the comfort of their living and dying was established; it was as though this thread which wavered up and filled their being, and yet which curved and wound back again to its source, suspended their speech, perhaps because it was a thin, golden ornament in space, perhaps because it brought to their minds for a few moments that they belonged to each other, and lifted them out of the dreadful stillness whose reverberations rise like an impenetrable wall of deafening silence between human being and human being, a wall through which the human voice cannot penetrate, so that it has to falter and die. (p.77)

There aren’t any real ‘ideas’ in this passage. Maybe it would be accurate to call it a kind of philosophically-minded description. A novel written by a philosophically-minded author. Not so much a novel of ideas as a novel of thoughtful descriptions.

The Romantic (1888) – plot summary, part one

So, I found the book quite difficult to get into because its style, layout, and approach are all alien to the super-accessible, Americanised prose we are all used to in 2020.

But, rather like getting into a cold swimming pool, if you persist, your body acclimatises to the style and you begin to grasp the basic structures of the novel, and on the back of that, to understand and appreciate what, after a while, you realise are moments of great beauty and sensitivity.

And you also come to realise that the book is built about sets of binary opposites with an almost mathematical precision (see my comment about schematic, above).

Joachim von Pasenow was sent by his landowning family to army cadet school aged ten (p.24), unlike his elder brother, Helmuth, who remained on the estate to run the family farm. The story opens with Joachim now aged about 30 (he has spent 20 years away from home, p.40), still in the army, a lieutenant (p.15) and about to be promoted to captain (p.89).

Town versus country Thus the brothers represent alternative destinies: Joachim lives in barracks in Berlin; Helmuth has stayed on the family farm in the country to help his ageing parents. So a basic binary in the novel is the contrast between urban people and values (‘you people who live in cities’) and rural lives and values (p.29).

One must not judge things merely from the standpoint of the city man; out there in the country people’s feelings were less artificial and meant more. (p.52)

But Joachim is not quite the representative of urban life I’ve just suggested. We soon get to know about a good friend of his who he met in the boy cadets and became a brother officer, Eduard von Bertrand. Bertrand quit the army and has become a businessman, a cotton importer (p.26), familiar with the Stock Exchange and the mysteries of banking and business ledgers. He has a

sureness and lightness of touch, and his competence in the affairs of life (p.147)

He has grown his hair curly (his ‘far too wavy hair’, p.51), wears smart suits, and has travelled widely, most recently to America, all qualities which Joachim mistrusts or actively despises.

So if Helmuth represents life back on the farm and Bertrand represents smart wheeler-dealer city life – then Joachim is the man in between – attracted but repelled by Bertrand’s stylish cynicism, equally attracted by his memories of simple life on the family farm, but repelled by the reality of his parents’ stultifying boredom and vulgarity.

The virgin versus the whore Same when it comes to women. Joachim’s father comes to see him in Berlin and the son is, reluctantly, obliged to take him to see the sights, which includes dinner at the Jäger Casino, where they come across two fancy women. (I think they’re high class prostitutes, but the social manners of the time being depicted and the elliptical way everyone refers to them don’t make it utterly clear.) Joachim’s father bluntly hands the dark-haired woman a 50-mark note with the apparent idea that he’s buying her for his son, but she suddenly runs off to cry with the lavatory attendant (?).

Joachim is, characteristically, disgusted by his father’s crudeness, but also haunted by the girl’s beauty and by the fleeting moment when she flirtatiously runs her hand over his close-cropped army haircut. His dad goes back to the farm, and Joachim spends days scouring the working class districts of Berlin with a half-formed intention to find the girl. One day she steps out of the crowd and into his life.

She is named Ruzena. She is not German but Czech, to be precise Bohemian (p.17), and speaks German badly (‘Not like you friend; he’s ugly man’) in a harsh staccato style.

Joachim takes Ruzena for lunch, then they take a carriage out to ‘the Havel’ – a park in West Berlin – she takes his arms under hers as they stroll beside the misty river, till it starts to rain and they take shelter under a tree where she leans against him. They kiss.

Back in Berlin he walks her to the door of her apartment where they kiss again, he turns and begins walking away, but turns again, runs up to her. She takes him upstairs and strips him and they have sex, for days afterwards he is haunted by the vision of her long black hair spread like a fan across her white pillow.

But – as usual – Joachim is conflicted. On his visit up to town, his father had suggested that Joachim pay court to the daughter of an aristocratic family in the neighbourhood, the Baddensens. She is named Elisabeth, who (to make things as simplistically symbolic as possible) is a posh, innocent blonde compared with Ruzena’s sensuous dark colouring. Elisabeth is the daughter of the Baron and Baroness von Baddensen, who live in the old manor-house on an estate at Lestow (p.24).

So Joachim is caught between the pure, angelic, blonde virgin of an eminent, rich family – or a raven-haired sex goddess, a courtesan who’s not even properly German, but has stolen his heart… or his loins, anyway.

Honour versus cynicism Yet another binary pairing occurs when Joachim’s elder brother is unexpectedly shot dead in a duel. a) In plot terms, since Joachim is now the only heir of the farm, his father wants Joachim to return to the land, to rural life with its illiterate peasants and simpler, Christian values. b) But in terms of the schema, Bertrand now grows in weight as a symbolic figure. He is given speeches praising city life, and deprecating rural values, especially rural – and by extension European – Christian faith.

In a striking speech he compares the waning of European Christianity with the passionate adherence of the African converts German and other European missionaries are making in Africa right now (1888). One day, Bertrand fancifully predicts, a great tidal wave of African Christians will sweep into a heathen Europe, reconverting it, and enthroning a black Pope in the Vatican (p.29).

The cause of Helmuth’s death wasn’t accidental. It was a duel, an old-fashioned duel, fought over ‘an affair of honour’ with a Polish landowner (though we never find out the precise cause).

So Helmuth’s straightforward ‘honour’ is compared & contrasted with Bertrand’s more worldly-wise cynicism. It’s not that Bertrand is a particularly fiery atheist, he is just a modern, successful business man who doesn’t understand how such 17th century values as duels and ‘honour’ have lived on into the age of trains and factories (p.51).

The character of Joachim von Pasenow

And, as usual, Joachim is the man in between, caught between his brother’s impeccable rectitude, which he himself feels was excessive, but repelled by Bertrand’s casual dismissal or at least questioning of it.

It is this aspect of being a man caught in between two worlds which really defines Joachim’s character, and the phrase ‘two worlds’ occurs a number of times in his internal monologues. He is perpetually uneasy.

During the last few days he had become uncertain about many things, and this in some inexplicable way was connected with Bertrand; some pillar or other of life had become shaky… and there had grown within him a longing for permanence, security and peace. (p.31)

Joachim, the man in the middle of all these binary opposites, could, I suppose have been wise and witty, or brisk and soldier-like – but instead he comes over as neurotic and tense, so profoundly confused, about even simple things like who he’s walking behind in the city streets, ‘so susceptible to this feeling of insecurity’, that the reader starts to think he must be having a nervous breakdown.

For a moment everything was confused again and one did not know to whom Ruzena belonged… (p.56)

Joachim is easily confused. He doesn’t understand other people’s motives, or over-thinks them. He confuses people in a surreal way, so that the sight of his fiancée Elisabeth climbing up into a train departing for the country is so exactly like the movements of his father undertaking the same action, that Joachim momentarily confuses them both, to such an extent that he becomes speechless.

‘In his fantasy’ (p.24), Joachim imagines Ruzena lives in one of the small shops he walks past in Berlin, with her dark-haired mother. All fantasy.

He sees an Italian-looking man at the Opera with black hair, hears him speaking a foreign language, and in a fantastical way comes to believe that it is Ruzena’s brother, on no evidence at all. He proceeds to superimpose her features on his, and the ‘brother’s’ features onto Ruzena – all baseless fantasia.

It is typical of Joachim’s diseased fantasy that, when he returns home for his brother’s funeral and sits in the room with the coffin he fantasises that it is he, Joachim, in the coffin (p.41). He dreams that Ruzena has killed herself by drowning herself in the river at the Havel Park – but next thing, is fantasising that it is he who has drowned, or that it is his eyes which look up out of her face (p.123).

A fantastic association led his thoughts quite into the absurd, and the confusion became almost inextricable… (p.88)

Walking through Berlin he finds himself following a fat bearded man waddling along and on absolutely no evidence concludes that he must be Bertrand’s business agent.

Even though Joachim knew that what he thought was without sense or sequence, yet it was as though the apparently confused skein concealed a sequence… (p.48)

He can’t control his neurotic and destructive fantasies.

After a while I noticed the number of sentences which include two perhapses: perhaps it was this, perhaps only that –

Perhaps they were tears he had not noticed, perhaps however it was only the oppressive heat… (p.44)

Well, that might be taken as sarcasm, or it might not (p.76)

Ambiguity and uncertainty are sewn into the fabric of the text throughout.

Joachim often gets confused by the actual experience of his thoughts. His thoughts hove into view but don’t quite crystallise or complete, before they melt away. His mind has many levels and on all of them he is subject to confused impressions, misidentifications, and ungraspable insights.

… at the same time and in some other layer of his mind… (p.35)

Then, just when it was becoming visible, the thought broke off and hid itself… (p.67)

A new feeling had unexpectedly risen in him; he tried to find words for it… (p.74)

Some of this feels a little like the interior monologue brought to unmatched heights in James Joyce’s Ulysses, but only a little. In Joyce’s novel entire passages are conveyed in a swirl of consciousness, in which language itself breaks down. Nothing like that happens here. Language remains correct and grammatical, it’s the characters thoughts which break down and evade their grasp.

Urban alienation The most obvious way that the book is ‘modernist’ is the way the central character’s confusion and neurosis is directly linked to the bustling crowds of late-nineteenth century Berlin (what the book describes as ‘the labyrinth of the city, p.22) which he finds overwhelming.

But now his thoughts jostled each other like the people in the crowd round about him, and even though he saw a goal in front of him which he wanted to reach, it swam and wavered and was lost to view like the back of the fat man before him. (p.49)

Against the anarchy of modern values Joachim the soldier struggles to hold himself erect and firm but is constantly fighting a losing battle.

It often required an actual effort to hold things firmly in their proper shapes, an effort to difficult that many a time all those people who bustled about as if all was in order seemed to him limited, blind and almost crazy. (p.113)

This is epitomised by the odd, extended passage early in the novel where Joachim tries to express to himself why the concept of the uniform is so important. For him his uniform is a ‘bulwark against anarchy’ (p.23) and the sight of civilian clothes sometimes makes Joachim feel physically sick.

The dangers of civilian life were of a more obscure and incomprehensible kind. Chaos and disorder everywhere, without a hierarchy, without discipline. (p.60)

When he meets Bertrand wearing civvies, Joachim is as embarrassed and ashamed to be seen with him as if he were naked (p.27). When his parents start sending him letters requesting that he quit the army and go to run the family farm, Joachim likens the idea to being stripped of his uniform and dumped naked in the Alexanderplatz (p.59).

The tangle of nets which stretched over the whole city, the net which he felt everywhere… an impenetrable, incomprehensible net of civilian values which was invisible and yet which darkened everything. (p.62)

Interlude: Why is the novel titled The Romantic?

It would be easy to answer that Joachim is a man whose head is full of ‘romantic’ notions of honour, duty, love and Christian faith and rural values, and the novel shows the stress all these ideas come under – but it’s not quite that simple.

For Joachim is far beyond having a ‘romantic’ turn of mind. He’s mad, actually. He regularly hallucinates – as in merging different people – is puzzled and confused about how to behave and what to think. And also he is simply too stupid to understand what Bertrand is saying half the time. I.e. Joachim is not a portrait of a throwback to an earlier, more romantic era – he is a neurotic on the edge of a breakdown, quite a lot more of a hard-edged figure.

Also he is a soldier. There’s a moment in Joachim’s rooms where Bertrand proposes an elaborate and humorous toast to Ruzena and, seeing it through Joachim’s eyes, we realise that he simply doesn’t understand what Bertrand is on about. He suspects it’s some complicated ploy to take Ruzena away from him, whereas the reader can see it’s just an elaborate and humorous toast.

Later in the book, Joachim tries to provide a regular income for Ruzena, and Bertrand recommends him to his lawyer to arrange it all, and the lawyer quickly sees that Joachim is useless at making decisions, in all the aspects of practical life.

Later still, in conversation with Elisabeth, Betrand tells her point blank that the ability to ‘love’ requires a modicum of wisdom, or at least cleverness – and that Joachim lacks both.

After a while I realised that Joachim is scared of everything and everyone. He is certain Bertrand is out to ‘get’ him, to drag him into civilian life, to steal his black-haired beauty or his blonde virgin. He is insistently paranoid. Unless his uniform is done up just so, unless he hold himself stiff and erect, then some nameless, dreadful thing will happen.

So it seems to me that Joachim is less a ‘romantic’ than a delusional, borderline hysterical, neurotic, extremely uptight and dim junior army officer. With the benefit of hindsight, we can see him as precisely the kind of narrow, patriotic, sexually tortured junior officer who went on to carry out countless coups throughout the 20th century, imprison and execute the liberal opposition, close bars and brothels and impose a strict sexual morality which reflects his own neuroses.

In conclusion, the protagonist of this novel is not at all what the title ‘The Romantic’ might lead you to believe.

Also, he isn’t the only ‘romantic’ character in the book. Elisabeth is, in her way, a desperate romantic i.e. she wants wishful fantasies to outweigh reality. She wants to live with her mummy and daddy forever and ever. And then there’s Ruzena who, Bertrand decides, is a romantic child, as helpless as a little animal.

So maybe the novel would more accurately be titled The Romantics.

The Romantic (1888) – plot summary, part two

Joachim is called down to his parents’ farm for the funeral of Helmuth. This means abandoning Ruzena in Berlin. She has just recently got a job as a showgirl-cum-actress through contacts of Bertrand’s.

Characteristically, Joachim had no idea about how to fulfil this ambition of hers ‘with all his mooning, romantic fantasies’ (p.64), whereas Bertrand was easily able to pull a few strings and make it happen. Which is why Joachim envies and despises him. (As the novel progresses we get more and more ‘leaks’ as to what Bertrand makes of his former comrade in arms; he thinks of Joachim as a ‘clumsy fellow’, p.92, and later on will simply call him stupid.)

Bertrand pays a courtesy call on Ruzena and walks her home and then Ruzena leans into him and lifts her mouth to be kissed, exactly as she did with Joachim. But Bertrand chastely kisses her cheek, she goes into her apartment block while he lights a cigar and strolls jauntily away. You begin to realise Bertrand has the measure of both Joachim and Ruzena, and is amusing himself with them.

Similarly, when Bertrand goes down to stay with the von Pasenow family at their estate in Stolpin, Joachim has a (characteristically) fatalistic intuition that Bertrand will take Elisabeth from him and, just as inevitably, Bertrand does.

The three go riding together and – in a strange and persuasive moment – Joachim reins his horse in just as it was about to take an easy jump, making it stumble and hurt its ankle; so that he reluctantly says he better walk it home – leaving Bertrand to embark on an extended and highly philosophical seduction of Elisabeth.

It is a characteristically Broch touch that Joachim doesn’t understand then or forever after just what impulse made him rein in his horse, thus almost certainly hurting it, thus forcing him to leave Bertrand and Elisabeth alone, thus almost certainly pushing them together, thus almost certainly sabotaging the plans the parents of both families have to make a convenient match between them.

It’s not rocket science, but it’s typical of Joachim’s puzzled personality that he agonises about it; and it’s typical of Broch’s approach to the novel, to the idea of fiction, that this is the kind of psychologically charged moment he likes to depict and then have his characters mull over for pages of dense, psychologically-charged prose.

Joachim’s father has a stroke. He begins behaving oddly. The stroke occurred when he was writing a furious letter disinheriting Joachim for his ‘treachery’ of insisting on going back to Berlin and refusing to stay and run the family farm. Joachim goes down to see him and stay. He pays some visits to Elisabeth where their relationship proceeds in a halting, frosty kind of way. After vegetating at the farm for some time, Joachim makes an excuse to return to Berlin for three days and immediately sends for Ruzena. She comes running, cooks for him, they go to bed. Joachim is unhappy with Ruzena’s career on the stage – where she gets plaudits from strange men – and suggests to her that he sets her up running a little lace shop.

This is a typically stupid Joachim suggestion based solely on the warm impression he gained from looking into a lace shop in which a mother and daughter were bent over their needles on one of his many walks around Berlin. Ruzena enjoys the attention she gets as a showgirl and so she angrily rejects Joachim’s suggestion, and angrily asks if he’s been put up to it by his ‘bad’ friend, Bertrand, who she’s never liked (p.117).

Joachim’s father deteriorates and so he is compelled to accompany a nerve specialist from Berlin down to the family home. Here the father makes another scene in a small gathering of his wife, Joachim, the village priest, the family doctor and the nerve specialist. He insists on rising from his sick bed and taking the head of the table from where he issues denunciations, telling everyone that his son Joachim is dead and buried in the local cemetery but still doesn’t write to him anymore. The people round the table look at each other. Father is losing his mind.

Meanwhile, Bertrand, back from a business trip to Prague, drops by Joachim’s flat to pay a courtesy call on Ruzena. Here he unwittingly presses all the wrong buttons, exacerbating her sense of grievance that Joachim wants to take her off the stage (and deny her the first really fulfilling activity she’s ever had in her life) and in a rather surprising development, she becomes so furious that she rummages around in Joachim’s drawers, finds his service revolver and shoots Bertrand.

Not badly. In the arm. She drops the revolver, he bleeds. It is a scene from an opera or a late 19th century melodrama. He insists she accompanies him in a hansom cab to the hospital where he has the wound dressed but when he comes out she has gone.

After seeking her in vain for a few days, Bertrand writes to Joachim who comes up from the farm. He explains what happened. Joachim sets out on a trawl of Berlin nightclubs, cafes etc. Eventually he finds Ruzena sprawled in the loos of a louche club. She is in a terrible state and has become a prostitute again. When he pleads with her to come home with him she locks herself in a cubicle. Joachim waits outside for an hour and then is horrified to see her emerging on the arm of a fat client, and getting into a cab together. Looks like his affair with the Bohemian beauty is over.

This leads to the sequence of scenes where Joachim, driven by ‘romantic’ notions, decides to settle some money on Ruzena. Bertrand’s lawyer sizes him up quickly, realising that the stiff-necked man in front of him is ‘helpless’ in the face of the real world (p.131). To Joachim, inside his head, everything feels tangled and entrapped in a closing mesh over which presides a vengeful God. Whereas to the lawyer facing him, Joachim’s case is one of a type he sees all the time – army officer of good family needs to pay off illicit lover, in order to clear way for marriage to eligible heiress, and he gives him brisk practical advice on how to do it, while useless Joachim sits in front of him, racked by terror of The Evil One.

Joachim goes straight from this meeting (stopping only to put on his best pair of army gloves) to the house in the western suburbs of the city which the von Banndensen family take for the season, knocks, enters, and asks Freiherr von Baddensen for his daughter’s hand in marriage. He and his wife are thrilled, but caution that they must speak to Elisabeth first.

A day or so later Joachim meets Bertrand and explains what he’s done. Bertrand is shrewd and supportive. In a classic piece of dramatic irony, the narrator then tells us why: that Elisabeth came to see Bertrand the day before, taking a carriage to the hospital and insisting on seeing him in a small private reception room to ask his advice.

Here they have a reprise of the semi-philosophical love-sparring which they had first had on the day of the horse ride. During this, Bertrand a) points out that Joachim is ‘too stupid to love’ (p.135) b) that he, Bertrand, loves Elisabeth, but will be leaving the country soon and so they cannot marry. Therefore c) she should marry Joachim.

It was difficult to gauge the tone of this. Is it light satire or – what it feels more like – Bertrand being quite brutally unfeeling and playing with Elisabeth’s emotions. All the time he is telling her they can’t be together, he is kissing her and telling her how much he loves her. Is he deliberately tormenting her? Or is he himself not quite in control of the situation? Anyway, having exhausted themselves, Elisabeth decides that she will marry Joachim and leaves the feverish Bertrand to return to his hospital bed.

The narration returns to the ‘now’ from which this flashback occurred i.e. to Joachim talking to Bertrand, and Joachim declares more fiercely than ever that he will marry Elisabeth.

This leads on to an extraordinary scene where Joachim pays a formal visit to the von Baddensens, there is a formal dinner, toasts are proposed in champagne, and then everyone leaves the happy couple alone. And there follows an extremely tense and embarrassing scene where the two lovers, neither of whom really wants to get married, have to go through the ghastly farce of Joachim getting down onto his knees to propose. In a very ‘modern’ touch, Joachim has a hallucination of the room’s walls moving away, of all the furniture moving away from him to an infinite distance while his heart freezes as he touches Elisabeth’s dainty little fingers which are as cold as ice (p.142), a chill which is like ‘a dreadful foreboding of death’ (p.153).

Not the least weird aspect of this very weird scene is that they both end up talking about Bertrand who is a more central part of their lives than each other.

In the coach back from the von Baddensens, Joachim has a typical one of his hallucinations, an overwhelming sense that both his father and Bertrand must have died, together, that evening. Of course, neither of them have. Joachim isn’t a ‘romantic’. He is delusional.

Joachim goes to see Bertrand in hospital and tell him about his proposal and acceptance and Bertrand is humorously supportive and, as always, Joachim feels he is being deluded, deceived, having rings run round him.

Elisabeth and Joachim get married. they fuss and fret about whether she’ll come to stay at his house, given that his father is now an invalid, or she go separately to stay with her parents, or whether they should go to the house in the suburbs of Berlin which her parents have gifted the couple. Joachim urgently needs the worldly wisdom of Bertrand to answer these questions for him, but Bertrand is not there.

During the marriage ceremony, Joachim is overcome with religious terror, that he is an imposter, one of the damned, and barely hears the words of the service at all. He is an Expressionist hysteric. He is screaming inside like Munch’s picture.

They go to stay in a hotel in Berlin and several pages are spent describing the inner turmoil of Joachim’s mind. In his head Elisabeth has always been a pure virginal figure – he is agonised by the presence a toilet next door, he cannot possibly imagine her using it – and he sees himself as her knight in shining armour devoted to protecting her. Thus the last few pages of the novel describe his agonising before he can bring himself to knock on her hotel door (they have separate rooms), going to the bed, kneeling beside it and kissing her hand. He wishes Bertrand were there to help him. He wishes Elisabeth were Ruzena with whom everything seemed easy and natural. By slow steps he lies down on the bed beside her and falls asleep. Elisabeth smiles and, after a while, falls asleep too.

This, I suppose, we are meant to take it, was the manner of the wooings and marriages of the Broch’s parents’ generation. Joachim is nuts, that’s extremely clear. And yet the message is subtler. For all the lies and evasions it is based on, I for one ended the book admiring the determination of both these dim, unprepared innocents to make the most of the situation they find themselves in. If they go on to have a formal, staid, distanced but affectionate and respectful marriage, who’s to say there’s anything wrong with that?

Religion

In the second half of the book, religion becomes a more and more dominant theme. Joachim’s confused thoughts gather together bits and pieces from the village priest, memories of extravagant religious pictures he saw in Dresden, attendance at church parades with his corps, and a few private visits to churches, to convince him that God is punishing him for his sins.

Inevitable fate, inescapable discipline of God! (p.122)

I can see how some readers might take this at face value and I’d be surprised if there aren’t hundreds of academic essay about religious imagery in the book. And yet to me it seems obvious that it’s all due to the fact that Joachim is an idiot.

He is terrified of civilians. He can’t handle the chaotic hustle and bustle of the big city. He doesn’t understand what Bertrand does, he doesn’t understand business. He has no idea how to make a bequest to Ruzena. He has no sense of how to run his parents’ farm as a business.

He is, in other words, hopeless and impractical and dim. His increasing turn to God and religion, therefore, seems to me the refuge of an idiot. Because he doesn’t understand anything about the actual world he finds himself in, he retreats to thinking it’s all part of a Divine Plan against him.

So, in my opinion, the religious aspect of the last third of the book has no real religious content but represents Joachim’s stupidity and his paranoia. It is more an investigation of how the stupid and the paranoid come to have religious faith. It’s not so much that it’s consoling (which it is) as that it is easy to understand. God is Daddy. Daddy is punishing me. I have been a bad boy. Not difficult, is it?

Descriptions

Once you have slowed right down to the speed of this odd book, and once you get into the habit of often rereading entire paragraphs to decipher what they’re about, I found myself admiring whole passages for their evocativeness and beauty.

They are not examples of good English prose, in fact they are often disfigured by unbeautiful phraseology (is that Broch or the Muirs?) but nonetheless there are passages of extended description which really manage to convey a room, a view, a landscape, a scene or setting and, in particular, the strange evanescent feeling of fleeting thoughts – with a depth and power which I found increasingly rewarding.

You really feel like you are entering the minds of the characters, above all the neurotic army lieutenant Joachim von Pasenow. Although, by the end, I wondered if the novel wasn’t about a so-called ‘romantic’ at all, but really about a near-simpleton. A good deal of Joachim’s agonising and tortured reflections about God or his uniform or civilian life etc really boil down to the fact that he’s a stupid person who doesn’t understand what’s going on around him, and finds it a real challenge coping with even the basics of adult life.

Maybe the book could have more accurately been titled The Idiot.

The dense crowd around him, the hubbub, as the Baroness called it, all this commercial turmoil full of faces and backs, seemed to him a soft, gliding, dissolving mass which one could not lay hold on. Where did it all lead to? (p.49)

Where indeed?

Credit

The English translation by Willa and Edwin Muir of The Sleepwalkers by Hermann Broch was first published in 1932. All references are to the Vintage International paperback edition of all three novels in one portmanteau volume which was first published in 1996.

20th century German literature

A Small Yes and a Big No by George Grosz (1946)

The Weimar Republic

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)

Cosmology

The Environment

Genetics and life

Human evolution

Maths

Particle physics

Atomic by Jim Baggott (2009)

Psychology

Nature’s Numbers by Ian Stewart (1995)

Ian Stewart is a mathematician and prolific author, having written over 40 books on all aspects of maths, as well as publishing several guides to the maths used in Terry Pratchett’s Discworld books, writing half a dozen textbooks for students, and co-authoring a couple of science fiction novels.

Stewart writes in a marvellously clear style but, more importantly, he is interesting: he sees the world in an interesting way, in a mathematical way, and manages to convey the wonder and strangeness and powerful insights which seeing the world in terms of patterns and shapes, numbers and maths, gives you.

He wants to help us see the world as a mathematician sees it, full of clues and information which can lead us to deeper and deeper appreciation of the patterns and harmonies all around us. It makes for a wonderfully illuminating read.

1. The Natural Order

Thus Stewart begins the book by describing just some of nature’s multitude of patterns: the regular movements of the stars in the night sky; the sixfold symmetry of snowflakes; the stripes of tigers and zebras; the recurring patterns of sand dunes; rainbows; the spiral of a snail’s shell; why nearly all flowers have petals arranged in one of the following numbers 5, 8, 13, 21, 34, 55, 89; the regular patterns or ‘rhythms’ made by animals scuttling, walking, flying and swimming.

2. What Mathematics is For

Mathematics is brilliant at helping us to solve puzzles. It is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity, and then using those rules and structures to explain what’s going on. (p.16)

Having gotten our attention, Stewart trots through the history of major mathematical discoveries including Kepler discovering that the planets move not in circles but in ellipses; the discovery that the nature of acceleration is ‘not a fundamental quality, but a rate of change’, then Newton and Leibniz inventing calculus to help us work outcomplex rates of change, and so on.

Two of the main things that maths are for are 1. providing the tools which let scientists understand what nature is doing 2. providing new theoretical questions for mathematicians to explore further. These are handy rules of thumb for distinguishing between, respectively, applied and pure mathematics.

Stewart mentions one of the oddities, paradoxes or thought-provoking things that crops up in many science books, which is the eerie way that good mathematics, mathematics well done, whatever its source and no matter how abstract its origin, eventually turns out to be useful, to be applicable to the real world, to explain some aspect of nature.

Many philosophers have wondered why. Is there a deep congruence between the human mind and the structure of the universe? Did God make the universe mathematically and implant an understanding of maths in us? Is the universe made of maths?

Stewart’s answer is simple and elegant: he thinks that nature exploits every pattern that there is, which is why we keep discovering patterns everywhere. We humans express these patterns in numbers, but nature doesn’t use numbers as such – she uses the patterns and shapes and possibilities which the numbers express or define.

Mendel noticing the numerical relationships with which characteristics of peas are expressed when they are crossbred. The double helix structure of DNA. Computer simulations of the evolution of the eye from an initial mutation creating a patch of skin cells sensitive to light, published by Daniel Nilsson and Susanne Pelger in 1994. Pattern appears wherever we look.

Resonance = the relationship between periodically moving bodies in which their cycles lock together so that they take up the same relative positions at regular intervals. The cycle time is the period of the system. The individual bodies have different periods. The moon’s rotational period is the same as its revolution around the earth, so there is a 1:1 resonance of its orbital and rotational periods.

Mathematics doesn’t just analyse, it can predict, predict how all kinds of systems will work, from the aerodynamics which keep planes flying, to the amount of fertiliser required to increase crop yield, to the complicated calculations which keep communications satellites in orbit round the earth and therefore sustain our internet and mobile phone networks.

Time lags The gap between a new mathematical idea being developed and its practical implementation can be a century or more: it was 17th century interest in the mathematics of vibrating violin strings which led, three hundred years later, to the invention of radio, radar and TV.

3. What Mathematics is About

The word ‘number’ does not have any immutable, God-given meaning. (p.42)

Numbers are the most prominent part of mathematics and everyone is taught arithmetic at school, but numbers are just one type of object that mathematics is interested in.

Stewart outlines the invention of whole numbers, and then of fractions. Some time in the Dark Ages the invention of 0. The invention of negative numbers, then of square roots. Irrational numbers. ‘Real’ numbers.

Whole numbers 1, 2, 3… are known as the natural numbers. If you include negative whole numbers, the series is known as integers. Positive and negative numbers taken together are known as rational numbers. Then there are real numbers and complex numbers. Five systems in total.

But maths is also about operations such as addition, subtraction, multiplication and division. And functions, also known as transformations, rules for transforming one mathematical object into another. Many of these processes can be thought of as things which help to create data structures.

Maths is like a landscape in which similar proofs and theories cluster together to create peaks and troughs.

4. The Constants of Change

Newton’s basic insight was that changes in nature can be described by mathematical processes. Stewart explains how detailed consideration of what happens to a cannonball fired out of a cannon helps us towards Newton’s fundamental law, that force = mass x acceleration.

Newton invented calculus to help work out solutions to moving bodies. Its two basic operations – integration and differentiation – mean that, given one element – force, mass or acceleration – you can work out the other two. Differentiation is the technique for finding rates of change; integration is the technique for ‘undoing’ the effect of differentiation in order to isolate out the initial variables.

Calculating rates of change is a crucial aspect of maths, engineering, cosmology and many other areas of science.

5. From Violins to Videos

He gives a fascinating historical recap of how initial investigations into the way a violin string vibrates gave rise to formulae and equations which turned out to be useful in mapping electricity and magnetism, which turned out to be aspects of the same fundamental force, electromagnetism. It was understanding this which underpinned the invention of radio, radar, TV etc and Stewart’s account describes the contributions made by Michael Faraday, James Clerk Maxwell, Heinrich Hertz and Guglielmo Marconi.

Stewart makes the point that mathematical theory tends to start with the simple and immediate and grow ever-more complicated. This is because of a basic approach common in lots of mathematics which is that, you have to start somewhere.

6. Broken Symmetry

A symmetry of an object or system is any transformation that leaves it invariant. (p.87)

There are many types of symmetry. The most important ones are reflections, rotations and translations.

7. The Rhythm of Life

The nature of oscillation and Hopf bifurcation (if a simplified system wobbles, then so must the complex system it is derived from) leads into a discussion of how animals – specifically animals with legs – move, which turns out to be by staggered or syncopated oscillations, oscillations of muscles triggered by neural circuits in the brain.

This is a subject Stewart has written about elsewhere and is something of an expert on. Thus he tells us that the seven types of quadrupedal gait are: the trot, pace, bound, walk, rotary gallop, transverse gallop, and canter.

8. Do Dice Play God?

This chapter covers Stewart’s take on chaos theory.

Chaotic behaviour obeys deterministic laws, but is so irregular that to the untrained eye it looks pretty much random. Chaos is not complicated, patternless behaviour; it is much more subtle. Chaos is apparently complicated, apparently patternless behaviour that actually has a simple, deterministic explanation. (p.130)

19th century scientists thought that, if you knew the starting conditions, and then the rules governing any system, you could completely predict the outcomes. In the 1970s and 80s it became increasingly clear that this was wrong. It is impossible because you can never define the starting conditions with complete certainty.

Thus all real world behaviours are subject to ‘sensitivity to initial conditions’. From minuscule divergences at the starting point, cataclysmic differences may eventually emerge in mature systems.

Stewart goes on to explain the concept of ‘phase space’ developed by Henri Poincaré: this is an imaginary mathematical space that represents all possible motions in a given dynamic system. The phase space is the 3-D place in which you plot the behaviour in order to create the phase portrait. Instead of having to define a formula and worrying about identifying every number of the behaviour, the general shape can be determined.

Much use of phase portraits has shown that dynamic systems tend to have set shapes which emerge and which systems move towards. These are called attractors.

9. Drops, Dynamics and Daisies

The book ends by drawing some philosophical conclusions.

Chaos theory has all sorts of implications but the one Stewart closes on is this: the world is not chaotic; if anything, it is boringly predictable. And at the level of basic physics and maths, the laws which seem to underpin it are also schematic and simple. And yet, what we are only really beginning to appreciate is how complicated things are in the middle.

It is as if nature can only get from simple laws (like Newton’s incredibly simple law of thermodynamics) to fairly simple outcomes (the orbit of the planets) via almost incomprehensibly complex processes.

To end, Stewart gives us three examples of the way apparently ‘simple’ phenomena in nature derive from stupefying complexity:

what exactly happens when a drop of water falls off a tap
computer modelling of the growth of fox and rabbit populations
why petals on flowers are arranged in numbers derived from the Fibonacci sequence

In all three cases the underlying principles seem to be resolvable into easily stated laws and functions – and in our everyday lives we see water dropping off taps or flowerheads all the time – and yet the intermediate steps between simple mathematical principles and real world embodiment turn out to be mind-bogglingly complex.

Coda: Morphomatics

Stewart ends the book with an epilogue speculating, hoping and wishing for a new kind of mathematics which incorporates chaos theory and the other elements he’s discussed – a theory and study of form, which takes everything we already know about mathematics and seeks to work out how the almost incomprehensible complexity we are discovering in nature gives rise to all the ‘simple’ patterns which we see around us. He calls it morphomatics.

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Reviews of other science books

Chemistry

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

Cosmology

The Environment

Genetics and life

Human evolution

Maths

Particle physics

Atomic by Jim Baggott (2009)

Psychology

2 Comments

by Simon on March 31, 2019 • Permalink

Posted in Books, Mathematics

Tagged 1995, addition, book, calculus, chaos theory, communications satellites, complex numbers, Daniel Nilsson and Susanne Pelger, data structures, differentiation, Discworld, division, DNA, double helix, earth, force = mass x acceleration, functions, Guglielmo Marconi, Heinrich Hertz, Henri Poincaré, Hopf bifurcation, Ian Stewart, integers, integration, James Clerk Maxwell, Kepler, laws, Leibniz, Mathematics, maths, Michael Faraday, morphomatics, multiplication, natural numbers, Nature, Nature's Numbers, negative numbers, Newton, operations, oscillation, patterns, period, phase portrait, phase space, rainbows; snail's shell, rational numbers, real numbers, reflection symmetry, resonance, rotation symmetry, rules, Science, square roots, strange attractors, subtraction, symmetry, Terry Pratchett, tigers, transformations, translation symmetry, zebras

Posted by Simon on March 31, 2019

https://astrofella.wordpress.com/2019/03/31/natures-numbers-ian-stewart/

Maths ideas from John Allen Paulos

There’s always enough random success to justify anything to someone who wants to believe.
(Innumeracy, p.33)

It’s easier and more natural to react emotionally than it is to deal dispassionately with statistics or, for that matter, with fractions, percentages and decimals.
(A Mathematician Reads the Newspaper p.81)

I’ve just read two of John Allen Paulos’s popular books about maths, A Mathematician Reads the Newspaper: Making Sense of the Numbers in the Headlines (1995) and Innumeracy: Mathematical Illiteracy and Its Consequences (1998).

My reviews tended to focus on the psychological, logical and cognitive errors which Paulos finds so distressingly common on modern TV and in newspapers, among politicians and commentators, and in every walk of life. I focused on these for the simple reason that I didn’t understand the way he explained most of his mathematical arguments.

I also criticised a bit the style and presentation of the books, which I found meandering, haphazard and so quite difficult to follow, specially since he was packing in so many difficult mathematical concepts.

Looking back at my reviews I realise I spent so much time complaining that I missed out promoting and explaining large chunks of the mathematical concepts he describes (sometimes at length, sometimes only in throwaway references).

This blog post is designed to give a list and definitions of the mathematical principles which John Allen Paulos describes and explains in these two books.

They concepts appear, in the list below, in the same order as they crop up in the books.

1. Innumeracy: Mathematical Illiteracy and Its Consequences (1988)

The multiplication principle If some choice can be made in M different ways and some subsequent choice can be made in B different ways, then there are M x N different ways the choices can be made in succession. If a woman has 5 blouses and 3 skirts she has 5 x 3 = 15 possible combinations. If I roll two dice, there are 6 x 6 = 36 possible combinations.

If, however, I want the second category to exclude the option which occurred in the first category, the second number is reduced by one. If I roll two dice, there are 6 x 6 = 36 possible combinations. But the number of outcomes where the number on the second die differs from the first one is 6 x 5. The number of outcomes where the faces of three dice differ is 6 x 5 x 4.

If two events are independent in the sense that the outcome of one event has no influence on the outcome of the other, then the probability that they will both occur is computed by calculating the probabilities of the individual events. The probability of getting two head sin two flips of a coin is ½ x ½ = ¼ which can be written (½)². The probability of five heads in a row is (½)⁵.

The probability that an event doesn’t occur is 1 minus the probability that it will occur. If there’s a 20% chance of rain, there’s an 80% chance it won’t rain. Since a 20% chance can also be expressed as 0.2, we can say there is a 0.2 chance it will rain and a 1 – 0.2 = 0.8 chance it won’t rain.

Binomial probability distribution arises whenever a procedure or trial may result in a ‘success’ or ‘failure’ and we are interested in the probability of obtaining R successes from N trials.

Dirichlet’s Box Principle aka the pigeonhole principle Given n boxes and m>n objects, at least one box must contain more than one object. If the postman has 21 letters to deliver to 20 addresses he knows that at least one address will get two letters.

Expected value The expected value of a quantity is the average of its values weighted according to their probabilities. If a quarter of the time a quantity equals 2, a third of the time it equals 6, another third of the time it equals 15, and the remaining twelfth of the time it equals 54, then its expected value is 12. (2 x ¼) + (6 x 1/3) + (15 x 1/3) + (54 x 1/12) = 12.

Conditional probability Unless the events A and B are independent, the probability of A is different from the probability of A given that B has occurred. If the event of interest is A and the event B is known or assumed to have occurred, ‘the conditional probability of A given B’, or ‘the probability of A under the condition B’, is usually written as P(A | B), or sometimes PB(A) or P(A / B).

For example, the probability that any given person has a cough on any given day may be only 5%. But if we know that the person has a cold, then they are much more likely to have a cough. The conditional probability of someone with a cold having a cough might be 75%. So the probability of any member of the public having a cough is 5%, but the probability of any member of the public who has a cold having a cough is 75%. P(Cough) = 5%; P(Cough | Sick) = 75%

The law of large numbers is a principle of probability according to which the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances. As the number of experiments increases, the actual ratio of outcomes will converge on the theoretical, or expected, ratio of outcomes.

For example, if a fair coin (where heads and tails come up equally often) is tossed 1,000,000 times, about half of the tosses will come up heads, and half will come up tails. The heads-to-tails ratio will be extremely close to 1:1. However, if the same coin is tossed only 10 times, the ratio will likely not be 1:1, and in fact might come out far different, say 3:7 or even 0:10.

The gambler’s fallacy a misunderstanding of probability: the mistaken belief that because a coin has come up heads a number of times in succession, it becomes more likely to come up tails. Over a very large number of instances the law of large numbers comes into play; but not in a handful.

Regression to the mean in any series with complex phenomena that are dependent on many variables, where chance is involved, extreme outcomes tend to be followed by more moderate ones. Or: the tendency for an extreme value of a random quantity whose values cluster around an average to be followed by a value closer to the average or mean.

Poisson probability distribution measures the probability that a certain number of events occur within a certain period of time. The events need to be a) unrelated to each other b) to occur with a known average rate. The Ppd can be used to work out things like the numbers of cars that pass on a certain road in a certain time, the number of telephone calls a call center receives per minute.

Bayes’ Theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes’ theorem, a person’s age can be used to more accurately assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person’s age.

Arrow’s impossibility theorem (1951) no rank-order electoral system can be designed that always satisfies these three “fairness” criteria:

If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
If every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y will also remain unchanged (even if voters’ preferences between other pairs like X and Z, Y and Z, or Z and W change).
There is no “dictator”: no single voter possesses the power to always determine the group’s preference.

The prisoner’s dilemma (1951) Two criminals are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. The prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The offer is:

If A and B each betray the other, each of them serves two years in prison
If A betrays B but B remains silent, A will be set free and B will serve three years in prison (and vice versa)
If A and B both remain silent, both of them will only serve one year in prison (on the lesser charge).

Prisoner’s dilemma graphic. Source: Wikipedia

Binomial probability Binomial means it has one of only two outcomes such as heads or tails. A binomial experiment is one that possesses the following properties:

The experiment consists of n repeated trials
Each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial)
The probability of a success, denoted by p, remains constant from trial to trial and repeated trials are independent.

The number of successes X in n trials of a binomial experiment is called a binomial random variable. The probability distribution of the random variable X is called a binomial distribution.

Type I and type II errors Type I error is where a true hypothesis is rejected. Type II error is where a false hypothesis is accepted.

Confidence interval Used in surveys, the confidence interval is a range of values, above and below a finding, in which the actual value is likely to fall. The confidence interval represents the accuracy or precision of an estimate.

Central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a “bell curve”) even if the original variables themselves are not normally distributed. OR: the sum or average of a large bunch of measurements follows a normal curve even if the individual measurements themselves do not. OR: averages and sums of non-normally distributed quantities will nevertheless themselves have a normal distribution. OR:

Under a wide variety of circumstances, averages (or sums) of even non-normally distributed quantities will nevertheless have a normal distribution (p.179)

Regression analysis here are many types of regression analysis, at their core they all examine the influence of one or more independent variables on a dependent variable. Performing a regression allows you to confidently determine which factors matter most, which factors can be ignored, and how these factors influence each other. In order to understand regression analysis you must comprehend the following terms:

Dependent Variable: This is the factor you’re trying to understand or predict.
Independent Variables: These are the factors that you hypothesize have an impact on your dependent variable.

Correlation is not causation a principle which cannot be repeated too often.

Gaussian distribution Gaussian distribution (also known as normal distribution) is a bell-shaped curve, and it is assumed that during any measurement values will follow a normal distribution with an equal number of measurements above and below the mean value.

The normal distribution is the most important probability distribution in statistics because it fits so many natural phenomena. For example, heights, blood pressure, measurement error, and IQ scores follow the normal distribution.

Statistical significance A result is statistically significant if it is sufficiently unlikely to have occurred by chance.

2. A Mathematician Reads the Newspaper: Making Sense of the Numbers in the Headlines

Incidence matrices In mathematics, an incidence matrix is a matrix that shows the relationship between two classes of objects. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each element of Y. The entry in row x and column y is 1 if x and y are related (called incident in this context) and 0 if they are not. Paulos creates an incidence matrix to show

Complexity horizon On the analogy of an ‘event horizon’ in physics, Paulos suggests this as the name for levels of complexity in society around us beyond which mathematics cannot go. Some things just are too complex to be understood using any mathematical tools.

Nonlinear complexity Complex systems often have nonlinear behavior, meaning they may respond in different ways to the same input depending on their state or context. In mathematics and physics, nonlinearity describes systems in which a change in the size of the input does not produce a proportional change in the size of the output.

The Banzhaf power index is a power index defined by the probability of changing an outcome of a vote where voting rights are not necessarily equally divided among the voters or shareholders. To calculate the power of a voter using the Banzhaf index, list all the winning coalitions, then count the critical voters. A critical voter is a voter who, if he changed his vote from yes to no, would cause the measure to fail. A voter’s power is measured as the fraction of all swing votes that he could cast. There are several algorithms for calculating the power index.

Vector field may be thought of as a rule f saying that ‘if an object is currently at a point x, it moves next to point f(x), then to point f(f(x)), and so on. The rule f is non-linear if the variables involved are squared or multiplied together and the sequence of the object’s positions is its trajectory.

Chaos theory (1960) is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.

‘Chaos’ is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions.

The butterfly effect describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, e.g. a butterfly flapping its wings in Brazil can cause a hurricane in Texas.

Linear models are used more often not because they are more accurate but because that are easier to handle mathematically.

All mathematical systems have limits, and even chaos theory cannot predict even relatively simple nonlinear situations.

Zipf’s Law states that given a large sample of words used, the frequency of any word is inversely proportional to its rank in the frequency table. So word number n has a frequency proportional to 1/n. Thus the most frequent word will occur about twice as often as the second most frequent word, three times as often as the third most frequent word, etc. For example, in one sample of words in the English language, the most frequently occurring word, “the”, accounts for nearly 7% of all the words (69,971 out of slightly over 1 million). True to Zipf’s Law, the second-place word “of” accounts for slightly over 3.5% of words (36,411 occurrences), followed by “and” (28,852). Only about 135 words are needed to account for half the sample of words in a large sample

Benchmark estimates Benchmark numbers are numbers against which other numbers or quantities can be estimated and compared. Benchmark numbers are usually multiples of 10 or 100.

Non standard models Almost everyone, mathematician or not, is comfortable with the standard model (N : +, ·) of arithmetic. Less familiar, even among logicians, are the non-standard models of arithmetic.

The S-curve A sigmoid function is a mathematical function having a characteristic “S”-shaped curve or sigmoid curve. Often, sigmoid function refers to the special case of the logistic function shown below

and defined by the formula:

This curve, sometimes called the logistic curve is extremely widespread: it appears to describe the growth of entities as disparate as Mozart’s symphony production, the rise of airline traffic, and the building of Gothic cathedrals (p.91)

Differential calculus The study of rates of change, rates of rates of change, and the relations among them.

Algorithm complexity gives on the length of the shortest program (algorithm) needed to generate a given sequence (p.123)

Chaitin’s theorem states that every computer, every formalisable system, and every human production is limited; there are always sequences that are too complex to be generated, outcomes too complex to be predicted, and events too dense to be compressed (p.124)

Simpson’s paradox (1951) A phenomenon in probability and statistics, in which a trend appears in several different groups of data but disappears or reverses when these groups are combined.

The amplification effect of repeated playing the same game, rolling the same dice, tossing the same coin.

Reviews of other science books

Chemistry

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

Cosmology

The Environment

Genetics and life

Human evolution

Maths

Particle physics

Atomic by Jim Baggott (2009)

Psychology

1 Comment

by Simon on March 30, 2019 • Permalink

Posted in Mathematics, Science

Tagged A Mathematician Reads the Newspaper: Making Sense of the Numbers in the Headlines, algorithm complexity, amplification effect, Arrow's impossibility theorem, Banzhaf power index, Bayes' Theorem, benchmark estimates, binomial probability, binomial probability distribution, Butterfly Effect, central limit theorem, Chaitin's theorem, chaos theory, coin toss, Complexity horizon, conditional probability, confidence interval, dependent variable, dice, differential calculus, Dirichlet's Box Principle, expected value, gambler's fallacy, Gaussian distribution, incidence matrices, independent variables, Innumeracy: Mathematical Illiteracy and Its Consequences, John Allen Paulos, law of large numbers, linear models, Mathematics, multiplication principle, non-standard models, nonlinear complexity, Poisson probability distribution, Prisoner's Dilemma, probability, regression analysis, regression to the mean, S-curve, sigmoid function, Simpson's paradox, statistical significance, statistics, type I error, type II error, Zipf's Law

Posted by Simon on March 30, 2019

https://astrofella.wordpress.com/2019/03/30/maths-ideas-from-john-allen-paulos/

A Mathematician Reads the Newspaper: Making Sense of the Numbers in the Headlines by John Allen Paulos (1995)

Always be smart. Seldom be certain. (p.201)

Mathematics is not primarily a matter of plugging numbers into formulas and performing rote computations. It is a way of thinking and questioning that may be unfamiliar to many of us, but is available to almost all of us. (p.3)

John Allen Paulos

John Allen Paulos is an American professor of mathematics who came to wider fame with publication of his short (130-page) primer, Innumeracy: Mathematical Illiteracy and its Consequences, published in 1988.

It was followed by Beyond Numeracy: Ruminations of a Numbers Man in 1991 and this book, A Mathematician Reads the Newspaper in 1995.

Structure

The book is made up of about 50 short chapters. He explains that each one of them will take a topic in the news in 1993 and 1994 and show how it can be analysed and understood better using mathematical tools.

The subjects of the essays are laid out under the same broad headings that you’d encounter in a newspaper, with big political stories at the front, giving way to:

Local, business and social issues
Lifestyle, spin and soft news
Science, medicine and the environment
Food, book reviews, sports and obituaries

Response

The book is disappointing in all kinds of ways.

First and foremost, he does not look at specific stories. All the headlines are invented. Each 4 or 5-page essay may or may not call in aspects of various topics in the news, but they do not look at one major news story and carefully deconstruct how it has been created and publicised in disregard of basic mathematics and probability and statistics. (This alone is highly suggestive of the possibility that, despite all his complaints to the contrary, specific newspaper stories where specific mathematical howlers are made and can be corrected are, in fact surprisingly rare.)

The second disappointment is that, even though these essays are very short, they cannot stay focused on one idea or example for much more than a page. I hate to say it and I don’t mean to be rude, but Paulos’s text has some kind of attention deficit disorder: the essays skitter all over the place, quickly losing whatever thread they ever had in a blizzard of references to politics, baseball, pseudoscience and a steady stream of bad jokes. He is so fond of digressions, inserts, afterthoughts and tangents that it is often difficult to say what any given essay is about.

I was hoping that each essay would take a specific news story and show how journalists had misunderstood the relevant data and maths to get it wrong, and would then show the correct way to analyse and interpret it. I was hoping that the 50 or so examples would have been carefully chosen to build up for the reader an armoury of techniques of arithmetic, probability, calculus, logarithms and whatever else is necessary to immediately spot, deconstruct and correct articles with bad maths in them.

Nope. Not at all.

Lani ‘Quota Queen’ Guinier

Take the very first piece, Lani ‘Quota Queen’ Guinier. For a start he doesn’t tell us who Lani ‘Quota Queen’ Guinier is. I deduce from his introduction that she was President Clinton’s nomination for the post of assistant attorney general for civil rights. We can guess, then, that the nickname ‘quota queen’ implies she was a proponent of quotas, though whether for black people, women or what is not explained.

Why not?

Paulos introduces us to the Banzhaf power index, devised in 1965 by lawyer John F. Banzhaf.

The Banzhaf power index of a group, party or person is defined to be the number of ways in which that group, party or person can change a losing coalition into a winning coalition or vice versa. (p.10)

He gives examples of companies where three or four shareholders hold different percentages of voting rights and shows how some coalitions of shareholders will always have decisive voting rights, whereas others never will (these are called the dummy) while even quite small shareholders can hold disproportionate power. For example in a situation where three shareholders hold 45%, 45% and 10% of the shares, the 10% party can often have the decisive say. In 45%, 45%, 8% and 2% the 2% is the dummy.

He then moves on to consider voting systems in some American states, including: cumulative voting, systems where votes don’t count as 1 but are proportionate to population, Borda counts (where voters rank the candidates and award progressively more points to those higher up the rankings), approval voting (where voters have as many votes as they want and can vote for as many candidates as they approve of), before going on to conclude that all voting systems have their drawbacks.

The essay ends with a typical afterthought, one-paragraph coda suggesting how the Supreme Court could end up being run by a cabal of just three judges. There are nine judges on the U.S. Supreme Court. Imagine (key word for Paulos), imagine a group of five judges agree to always discuss issues among themselves first, before the vote of the entire nine, and imagine they decide to always vote according to whatever the majority (3) decide. Then imagine that a sub-group of just three judges go away and secretly decide, that in the group of five, they will always agree. Thus they will dictate the outcome of every Supreme Court decision.

So:

1. I had no idea who Lani ‘Quota Queen’ Guinier was or, more precisely, I had to do a bit of detective work to figure it out, and still wasn’t utterly sure.

2. This is a very sketchy introduction to the issue of democratic voting systems. This is a vast subject, which Paulos skates over quickly and thinly.

Electoral systems

Thus, in these four and a bit pages you have the characteristic Paulos experience of feeling you are wandering all over the place, not quite at random, but certainly not in a carefully planned sequential way designed to explore a topic thoroughly and reach a conclusion. You are introduced to a number of interesting ideas, with some maths formulae, but not in enough detail or at sufficient length to really understand them. And because he’s not addressing any particular newspaper report or article, there are no particular misconceptions to clear up: the essay is a brief musing, a corralling of thoughts on an interesting topic.

This scattergun approach characterises the whole book.

Psychological availability and anchoring effects

The second essay is titled Psychological availability and anchoring effects. He explains what the availability error, the anchor effect and the halo effect are. If this is the first time you’ve come across these notions, they’re powerful new ideas. But I recently reread Irrationality by Stuart Sutherland which came out three years before Paulos’s book and spends over three hundred pages investigating these and all the other cognitive biases which afflict mankind in vastly more depth than Paulos, with many more examples. Next to it, Paulos’s three-minute essay seemed sketchy and superficial.

General points

Rather than take all 50 essays to pieces, here are notes on what I actually did learn. Note that almost none of it was about maths, but general-purpose cautions about how the news media work, and how to counter its errors of logic. In fact, all of it could have come from a media studies course without any maths at all:

almost all ‘news’ reinforces conventional wisdom
because they’re so brief, almost all headlines must rely on readers’ existing assumptions and prejudices
almost all news stories relate something new back to similar examples from the past, even when the comparison is inappropriate, again reinforcing conventional wisdom and failing to recognise the genuinely new

all economic forecasts are rubbish: this is because economics (like the weather and many other aspects of everyday life) is a non-linear system. Chaos theory shows that non-linear systems are highly sensitive to even minuscule differences in starting conditions, which has been translated into pop culture as the Butterfly Effect
and also with ‘futurologists’: the further ahead they look, the less reliable their predictions

the news is deeply biased by always assuming human agency is at work in any outcome: if any disaster happens anywhere the newspapers always go searching for a culprit; in the present Brexit crisis lots of news outlets are agreeing to blame Theresa May. But often things happen at random or as an accumulation of unpredictable factors. Humans are not good at acknowledging the role of chance and randomness.

There is a tendency to look primarily for culpability and conflicts of human will rather than at the dynamics of a natural process. (p.160)

Hence so many newspapers endlessly playing the blame game. The Grenfell Tower disaster was, first and foremost, an accident in the literal sense of ‘an unfortunate incident that happens unexpectedly and unintentionally, typically resulting in damage or injury’ – but you won’t find anybody who doesn’t fall in with the prevailing view that someone must be to blame. There is always someone to blame. We live in a Blame Society.
personalising beats stats, data or probability: nothing beats ‘the power of dramatic anecdote’ among the innumerate: ‘we all tend to be unduly swayed by the dramatic, the graphic, the visceral’ (p.82)
if you combine human beings’ tendency to personalise everything, and to look for someone to blame, you come up with Donald Trump, who dominates every day’s news

so much is happening all the time, in a world with more people and incidents than ever before, in which we are bombarded with more information via more media than ever before – that it would be extraordinary if all manner or extraordinary coincidences, correspondences and correlations didn’t happen all the time
random events can sometimes present a surprisingly ordered appearance
because people imbue meaning into absolutely everything, then the huge number of coincidences and correlations are wrongfully interpreted as meaningful

Tips and advice

I was dismayed at the poor quality of many of the little warnings which each chapter ends with. Although Paulos warns against truisms (on page 54) his book is full of them.

Local is not what it used to be, and we shouldn’t be surprised at how closely we’re linked. (p.55)

In the public realm, often the best we can do is to stand by and see how events unfold. (p.125)

Chapter three warns us that predictions about complex systems (the weather, the economy, big wars) are likely to be more reliable the simpler the system they’re predicting, and the shorter period they cover. Later he says we should be sceptical about all long-term predictions by politicians, economists and generals.

It didn’t need a mathematician to tell us that.

A lot of it just sounds like a grumpy old man complaining about society going to the dogs:

Our increasingly integrated and regimented society undermines our sense of self… Meaningless juxtapositions and coincidences replace conventional narratives and contribute to our dissociation… (pp.110-111)

News reports in general, and celebrity coverage in particular, are becoming ever-more self-referential. (p.113)

We need look no further than the perennial appeal of pseudoscientific garbage, now being presented in increasingly mainstream forums… (p.145)

The fashion pages have always puzzled me. In my smugly ignorant view, they appear to be so full of fluff and nonsense as to make the astrology columns insightful by comparison. (p.173)

Another aspect of articles in the society pages or in the stories about political and entertainment figures is the suggestion that ‘everybody’ knows everybody else. (p.189)

Sometimes his liberal earnestness topples into self-help book touchy-feeliness.

Achieving personal integration and a sense of self is for the benefit of ourselves and those we’re close to. (p.112)

But just occasionally he does say something unexpected:

The attention span created by television isn’t short; it’s long, but very, very shallow. (p.27)

That struck me as an interesting insight but, as with all his interesting comments, no maths was involved. You or I could have come up with it from general observation.

Complexity horizon

The notion that the interaction of human laws, conventions, events, politics, and general information overlap and interplay at ever-increasing speeds to eventually produce situations so complex as to appear unfathomable. Individuals, and groups and societies, have limits of complexity beyond which they cannot cope, but have to stand back and watch. Reading this made me think of Brexit.

He doesn’t mention it, but a logical spin-off would be that every individual has a complexity quotient like an intelligence quotient or IQ. Everyone could take a test in which they are faced with situations of slowly increasing complexity – or presented with increasingly complex sets of information – to find out where their understanding breaks off – which would become their CQ.

Social history

The book was published in 1995 and refers back to stories current in the news in 1993 and 1994. The run of domestic political subjects he covers in the book’s second quarter powerfully support my repeated conviction that it is surprising how little some issues have changed, how little movement there has been on them, and how they have just become a settled steady part of the social landscape of our era.

Thus Paulos has essays on:

gender bias in hiring
homophobia
accusations of racism arising from lack of ethnic minorities in top jobs (the problem of race crops up numerous times (pp.59-62, p.118)
the decline in educational standards
the appallingly high incidence of gun deaths, especially in black and minority communities
the fight over abortion

I feel increasingly disconnected from contemporary politics, not because it is addressing new issues I don’t understand, but for the opposite reason: it seems to be banging on about the same issues which I found old and tiresome twenty-five years ago.

The one topic which stood out as having changed is AIDS. In Innumeracy and in this book he mentions the prevalence or infection rates of AIDS and is obviously responding to numerous news stories which, he takes it for granted, report it in scary and alarmist terms. Reading these repeated references to AIDS made me realise how completely and utterly it has fallen off the news radar in the past decade or so.

In the section about political correctness he makes several good anti-PC points:

democracy is about individuals, the notion that everyone votes according to their conscience and best judgement; as soon as you start making it about groups (Muslims, blacks, women, gays) you start undermining democracy
racism and sexism and homophobia are common enough already without making them the standard go-to explanations for social phenomena which often have more complex causes; continually attributing all aspects of society to just a handful of inflammatory issues, keeps the issues inflammatory
members of groups often vie with each other to assert their loyalty, to proclaim their commitment to the party line and this suggests a powerful idea: that the more opinions are expressed, the more extreme these opinions will tend to become. This is a very relevant idea to our times when the ubiquity of social media has a) brought about a wonderful spirit of harmony and consensus, or b) divided society into evermore polarised and angry groupings

Something bad is coming

I learned to fear several phrases which indicate that a long, possibly incomprehensible and frivolously hypothetical example is about to appear:

‘Imagine…’

Imagine flipping a penny one thousand times in succession and obtaining some sequence of heads and tails… (p.75)

Imagine a supercomputer, the Delphic-Cray 1A, into which has been programmed the most complete and up-to-date scientific knowledge, the initial condition of all particles, and sophisticated mathematical techniques and formulas. Assume further that… Let’s assume for argument’s sake that… (p.115)

Imagine if a computer were able to generate a random sequence S more complex than itself. (p.124)

Imagine the toast moistened, folded, and compressed into a cubical piece of white dough… (p.174)

Imagine a factory that produces, say, diet food. Let’s suppose that it is run by a sadistic nutritionist… (p.179)

‘Assume that…’

Let’s assume that each of these sequences is a billion bits long… (p.121)

Assume the earth’s oceans contain pristinely pure water… (p.141)

Assume that there are three competing healthcare proposals before the senate… (p.155)

Assume that the probability of your winning the coin flip, thereby obtaining one point, is 25 percent. (p.177)

Assume that these packages come off the assembly line in random order and are packed in boxes of thirty-six. (p.179)

Jokes and Yanks

All the examples are taken from American politics (President Clinton), sports (baseball) and wars (Vietnam, First Gulf War) and from precisely 25 years ago (on page 77, he says he is writing in March 1994), both of which emphasise the sense of disconnect and irrelevance with a British reader in 2019.

As my kids know, I love corny, bad old jokes. But not as bad as the ones the book is littered with:

And then there was the man who answered a matchmaking company’s computerised personals ad in the paper. He expressed his desire for a partner who enjoys company, is comfortable in formal wear, likes winter sports, and is very short. The company matched him with a penguin. (pp.43-44)

The moronic inferno and the liberal fallacy

The net effect of reading this book carefully is something that the average person on the street knew long ago: don’t believe anything you read in the papers.

And especially don’t believe any story in a newspaper which involves numbers, statistics, percentages, data or probabilities. It will always be wrong.

More broadly his book simply fails to take account of the fact that most people are stupid and can’t think straight, even very, very educated people. All the bankers whose collective efforts brought about the 2008 crash. All the diplomats, strategists and military authorities who supported the Iraq War. All the well-meaning liberals who supported the Arab Spring in Egypt and Libya and Syria. Everyone who voted Trump. Everyone who voted Brexit.

Most books of this genre predicate readers who are white, university-educated, liberal middle class and interested in news and current affairs, the arts etc and – in my opinion – grotesquely over-estimate both their value and their relevance to the rest of the population. Because this section of the population – the liberal, university-educated elite – is demonstrably in a minority.

Over half of Americans believe in ghosts, and a similar number believes in alien abductions. A third of Americans believe the earth is flat, and that the theory of evolution is a lie. About a fifth of British adults are functionally illiterate and innumerate. This is what Saul Bellow referred to as ‘the moronic inferno’.

On a recent Radio 4 documentary about Brexit, one contributor who worked in David Cameron’s Number Ten commented that he and colleagues went out to do focus groups around the country to ask people whether we should leave the EU and that most people didn’t know what they were talking about. Many people they spoke to had never heard of the European Union.

On page 175 he says the purpose of reading a newspaper is to stretch the mind, to help us envision distant events, different people and unusual situations, and broaden our mental landscape.

Is that really why he thinks people read newspapers? As opposed to checking the sports results, catching up with celebrity gossip, checking what’s happening in the soaps, reading interviews with movie and pop stars, looking at fashion spreads, reading about health fads and, if you’re one of the minority who bother with political news, having all your prejudices about how wicked and stupid the government, the poor, the rich or foreigners etc are, and despising everyone who disagrees with you (Guardian readers hating Daily Mail readers; Daily Mail readers hating Guardian readers; Times readers feeling smugly superior to both).

This is a fairly entertaining, if very dated, book – although all the genuinely useful bits are generalisations about human nature which could have come from any media studies course.

But if it was intended as any kind of attempt to tackle the illogical thinking and profound innumeracy of Western societies, it is pissing in the wind. The problem is vastly bigger than this chatty, scattergun and occasionally impenetrable book can hope to scratch. On page 165 he says that a proper understanding of mathematics is vital to the creation of ‘an informed and effective citizenry’.

‘An informed and effective citizenry’?

Reviews of other science books

Chemistry

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

Cosmology

The Environment

Genetics and life

Human evolution

Maths

Particle physics

Atomic by Jim Baggott (2009)

Psychology

1 Comment

by Simon on March 26, 2019 • Permalink

Posted in America, Books, Mathematics, Science

Posted by Simon on March 26, 2019

https://astrofella.wordpress.com/2019/03/26/a-mathematician-reads-the-newspaper-john-allen-paulos/

Innumeracy by John Allen Paulos (1988)

Our innate desire for meaning and pattern can lead us astray… (p.81)

Giving due weight to the fortuitous nature of the world is, I think, a mark of maturity and balance. (p.133)

John Allen Paulos is an American professor of mathematics who won fame beyond his academic milieu with the publication of this short (134-page) but devastating book thirty years ago, the first of a series of books popularising mathematics in a range of spheres from playing the stock market to humour.

As Paulos explains in the introduction, the world is full of humanities graduates who blow a fuse if you misuse ‘infer’ and ‘imply’, or end a sentence with a dangling participle, but are quite happy to believe and repeat the most hair-raising errors in maths, statistics and probability.

The aim of this book was:

to lay out examples of classic maths howlers and correct them
to teach readers to be more alert when maths, stats and data need to be used
and to provide basic rules in order to understand when innumerate journalists, politicians, tax advisors and other crooks are trying to pull the wool over your eyes, or are just plain wrong

There are five chapters:

Examples and principles
Probability and coincidence
Pseudoscience
Whence innumeracy
Statistics, trade-offs and society

Many common themes emerge:

Don’t personalise, numeratise

One contention of this book is that innumerate people characteristically have a strong tendency to personalise – to be misled by their own experiences, or by the media’s focus on individuals and drama… (p.1)

Powers

The first chapter uses lots of staggering statistics to get the reader used to very big and very small numbers, and how to compute them.

1 million seconds is 11 and a half days. 1 billion seconds is 32 years.

He suggests you come up with personal examples of numbers for each power up to 12 or 13 i.e. meaningful embodiments of thousands, tens of thousands, hundreds of thousands and so on to help you remember and contextualise them in a hurry.

A snail moves at 0.005 miles an hour, Concorde at 2,000 miles per hour. Escape velocity from earth is about 7 miles per second, or 25,000 miles per hour. The mass of the Earth is 5.98 x 10²⁴ kg

Early on he tells us to get used to the nomenclature of ‘powers’ – using 10 to the power 3 or 10³ instead of 1,000, or 10 to negative powers to express numbers below 1. (In fact, right at this early stage I found myself stumbling because one thousand means more to me that 10³ and a thousandth means more than more 10^-3 but if you keep at it, it is a trick you can acquire quite quickly.)

The additive principle

He introduces us to basic ideas like the additive principle (aka the rule of sum), which states that if some choice can be made in M different ways and some subsequent choice can be made in N different ways, then there are M x N different ways these choices can be made in succession – which can be applied to combinations of multiple items of clothes, combinations of dishes on a menu, and so on.

Thus the number of results you get from rolling a die is 6. If you roll two dice, you can now get 6 x 6 = 36 possible numbers. Three numbers = 216. If you want to exclude the number you get on the first dice from the second one, the chances of rolling two different numbers on two dice is 6 x 5, of rolling different numbers on three dice is 6 x 5 x 4, and so on.

Thus: Baskin Robbins advertises 31 different flavours of ice cream. Say you want a triple scoop cone. If you’re happy to have any combination of flavours, including where any 2 or 3 flavours are the same – that’s 31 x 31 x 31 = 29,791. But if you ask how many combinations of flavours there are, without a repetition of the same flavour in any of the cones – that is 31 x 30 x 29 = 26,970 ways of combining.

Probability

I struggled with even the basics of probability. I understand a 1 in five chance of something happening, reasonably understand a 20% chance of something happening, but struggled when probability was expressed as a decimal number e.g. 0.2 as a way of writing a 20 percent or 1 in 5 chance.

With the result that he lost me on page 16 on or about the place where he explained the following example.

Apparently a noted 17th century gambler asked the famous mathematician Pascal which is more likely to occur: obtaining at least one 6 in four rolls of a single die, or obtaining at least one 12 in twenty four rolls of a pair of dice. Here’s the solution:

Since 5/6 is the probability of not rolling a 6 on a single roll of a die, (5/6)⁴is the probability of not rolling a 6 in four rolls of the die. Subtracting this number from 1 gives us the probability that this latter event (no 6s) doesn’t occur; in other words, of there being at least one 6 rolled in four tries: 1 – (5/6)⁴= .52. Likewise, the probability of rolling at least one 12 in twenty-four rolls of a pair of dice is seen to be 1 – (35/36)²⁴ = .49.

a) He loses me in the second sentence which I’ve read half a dozen times and still don’t understand – it’s where he says the chances that this latter event doesn’t occur: something about the phrasing there, about the double negative, loses me completely, with the result that b) I have no idea whether .52 is more likely or less likely than .49.

He goes on to give another example: if 20% of drinks dispensed by a vending machine overflow their cups, what is the probability that exactly three of the next ten will overflow?

The probability that the first three drinks overflow and the next seven do not is (.2)³x (.8)⁷. But there are many different ways for exactly three of the ten cups to overflow, each way having probability (.2)³x (.8)⁷. It may be that only the last three cups overflow, or only the fourth, fifth and ninth cups, and so on. Thus, since there are altogether (10 x 9 x 8) / (3 x 2 x 1) = 120 ways for us to pick three out of the ten cups, the probability of some collection of exactly three cups overflowing is 120 x (.2)³x (.8)^7.

I didn’t understand the need for the (10 x 9 x 8) / (3 x 2 x 1) equation – I didn’t understand what it was doing, and so didn’t understand what it was measuring, and so didn’t understand the final equation. I didn’t really have a clue what was going on.

In fact, by page 20, he’d done such a good job of bamboozling me with examples like this that I sadly concluded that I must be innumerate.

More than that, I appear to have ‘maths anxiety’ because I began to feel physically unwell as I read that problem paragraph again and again and again and didn’t understand it. I began to feel a tightening of my chest and a choking sensation in my throat. Rereading it now is making it feel like someone is trying to strangle me.

Maybe people don’t like maths because being forced to confront something you don’t understand, but which everyone around you is saying is easy-peasy, makes you feel ill.

Maths anxiety

2. Probability and coincidence

Having more or less given up on trying to understand Paulos’s maths demonstrations in the first twenty pages, I can at least latch on to his verbal explanations of what he’s driving at, in sentences like these:

A tendency to drastically underestimate the frequency of coincidences is a prime characteristic of innumerates, who generally accord great significance to correspondences of all sorts while attributing too little significance to quite conclusive but less flashy statistical evidence. (p.22)

It would be very unlikely for unlikely events not to occur. (p.24)

There is a strong general tendency to filter out the bad and the failed and to focus on the good and the successful. (p.29)

Belief in the… significance of coincidences is a psychological remnant of our past. It constitutes a kind of psychological illusion to which innumerate people are particularly prone. (p.82)

Slot machines light up and make a racket when people win, there is unnoticed silence for all the failures. Big winners on the lottery are widely publicised, whereas every one of the tens of millions of failures is not.

One result is ‘Golden Age’ thinking when people denigrate today’s sports or arts or political figures, by comparison with one or two super-notable figures from the vast past, Churchill or Shakespeare or Michelangelo, obviously neglecting the fact that there were millions of also-rans and losers in their time as well as ours.

The Expected value of a quality is the average of its values weighted according to their probabilities. I understood these words but I didn’t understand any of the five examples he gave.

The likelihood of probability In many situations, improbability is to be expected. The probability of being dealt a particular hand of 13 cards in bridge is less than 1 in 600 billion. And yet it happens every time someone is dealt a hand in bridge. The improbable can happen. In fact it happens all the time.

The gambler’s fallacy The belief that, because a tossed coin has come up tails for a number of tosses in a row, it becomes steadily more likely that the next toss will be a head.

3. Pseudoscience

Paulos rips into Freudianism and Marxism for the way they can explain away any result counter to their ‘theories’. The patient gets better due to therapy: therapy works. The patient doesn’t get better during therapy, well the patient was resisting, projecting their neuroses on the therapist, any of hundreds of excuses.

But this is just warming up before he rips into a real bugbear of his, the wrong-headedness of Parapsychology, the Paranormal, Predictive dreams, Astrology, UFOs, Pseudoscience and so on.

As with predictive dreams, winning the lottery or miracle cures, many of these practices continue to flourish because it’s the handful of successes which stand out and grab our attention and not the thousands of negatives.

Probability

As Paulos steams on with examples from tossing coins, rolling dice, playing roulette, or poker, or blackjack, I realise all of them are to do with probability or conditional probability, none of which I understand.

This is why I have never gambled on anything, and can’t play poker. When he explains precisely how accumulating probabilities can help you win at blackjack in a casino, I switch off. I’ve never been to a casino. I don’t play blackjack. I have no intention of ever playing blackjack.

When he says that probability theory began with gambling problems in the seventeenth century, I think, well since I don’t gamble at all, on anything, maybe that’s why so much of this book is gibberish to me.

Medical testing and screening

Apart from gambling the two most ‘real world’ areas where probability is important appear to be medicine and risk and safety assessment. Here’s an extended example he gives of how even doctors make mistakes in the odds.

Assume there is a test for cancer which is 98% accurate i.e. if someone has cancer, the test will be positive 98 percent of the time, and if one doesn’t have it, the test will be negative 98 percent of the time. Assume further that .5 percent – one out of two hundred people – actually have cancer. Now imagine that you’ve taken the test and that your doctor sombrely informs you that you have tested positive. How depressed should you be? The surprising answer is that you should be cautiously optimistic. To find out why, let’s look at the conditional probability of your having cancer, given that you’ve tested positive.

Imagine that 10,000 tests for cancer are administered. Of these, how many are positive? On the average, 50 of these 10,000 people (.5 percent of 10,000) will have cancer, and, so, since 98 percent of them will test positive, we will have 49 positive tests. Of the 9,950 cancerless people, 2 percent of them will test positive, for a total of 199 positive tests (.02 x 9,950 = 199). Thus, of the total of 248 positive tests (199 + 49 = 248), most (199) are false positives, and so the conditional probability of having cancer given that one tests positive is only 49/248, or about 20 percent! (p.64)

I struggled to understand this explanation. I read it four or five times, controlling my sense of panic and did, eventually, I think, follow the argumen.

However, worse in a way, when I think I did finally understand it, I realised I just didn’t care. It’s not just that the examples he gives are hard to follow. It’s that they’re hard to care about.

Whereas his descriptions of human psychology and cognitive errors in human thinking are crystal clear and easy to assimilate:

If we have no direct evidence of theoretical support for a story, we find that detail and vividness vary inversely with likelihood; the more vivid details there are to a story, the less likely the story is to be true. (p.84)

4. Whence innumeracy?

It came as a vast relief when Paulos stopped trying to explain probability and switched to a long chapter puzzling over why innumeracy is so widespread in society, which kicks off by criticising the poor level of teaching of maths in school and university.

This was like the kind of hand-wringing newspaper article you can read any day of the week in a newspaper or online, and so felt reassuringly familiar and easy to assimilate. I stopped feeling so panic-stricken.

This puzzling over the disappointing level of innumeracy goes on for quite a while. Eventually it ends with a digression about what appears to be a pet idea of his: the notion of introducing a safety index for activities and illnesses.

Paulos’s suggestion is that his safety index would be on a logarithmic scale, like the Richter Scale – so straightaway he has to explain what a logarithm is: The logarithm for 100 is 2 because 100 is 10², the logarithm for 1,000 is 3 because 1,000 is 10³. I’m with him so far, as he goes on to explain that the logarithm of 700 i.e. between 2 (100) and 3 (1,000) is 2.8. Since 1 in 5,300 Americans die in a car crash each year, the safety index for driving would be 3.7, the logarithm of 5,300. And so on with numerous more examples, whose relative risks or dangers he reduces to figures like 4.3 and 7.1.

I did understand his aim and the maths of this. I just thought it was bonkers:

1. What is the point of introducing a universal index which you would have to explain every time anyone wanted to use it? Either it is designed to be usable by the widest possible number of citizens; or it is a neat exercise on maths to please other mathematicians and statisticians.

2. And here’s the bigger objection – What Paulos, like most of the university-educated, white, liberal intellectuals I read in papers, magazines and books, fails to take into account is that a large proportion of the population is thick.

Up to a fifth of the adult population of the UK is functionally innumerate, that means they don’t know what a ‘25% off’ sign means on a shop window. For me an actual social catastrophe being brought about by this attitude is the introduction of Universal Credit by the Conservative government which, from top to bottom, is designed by middle-class, highly educated people who’ve all got internet accounts and countless apps on their smartphones, and who have shown a breath-taking ignorance about what life is like for the poor, sick, disabled, illiterate and innumerate people who are precisely the people the system is targeted at.

Same with Paulos’s scheme. Smoking is one of the most dangerous and stupid things which any human can do. Packs of cigarettes have for years, now, carried pictures of disgusting cancerous growths and the words SMOKING KILLS. And yet despite this, about a fifth of adults, getting on for 10 million people, still smoke. 🙂

Do you really think that introducing a system using ornate logarithms will get people to make rational assessments of the risks of common activities and habits?

Paulos then goes on to complicate the idea by suggesting that, since the media is always more interested in danger than safety, maybe it would be more effective, instead of creating a safety index, to create a danger index.

You would do this by

working out the risk of an activity (i.e. number of deaths or accidents per person doing the activity)
converting that into a logarithmic value (just to make sure than nobody understands it) and then
subtracting the logarithmic value of the safety index from 10, in order to create a danger index

He goes on to say that driving a car and smoking would have ‘danger indices’ of 3.7 and 2.9, respectively. The trouble was that by this point I had completely ceased to understand what he’s saying. I felt like I’ve stepped off the edge of a tall building into thin air. I began to have that familiar choking sensation, as if someone was squeezing my chest. Maths anxiety.

Under this system being kidnapped would have a safety index of 6.7. Playing Russian roulette once a year would have a safety index of 0.8.

It is symptomatic of the uselessness of the whole idea that Paulos has to remind you what the values mean (‘Remember that the bigger the number, the smaller the risk.’ Really? You expect people to run with this idea?)

Having completed the danger index idea, Paulos returns to his extended lament on why people don’t like maths. He gives a long list of reasons why he thinks people are so innumerate a condition which is, for him, a puzzling mystery.

For me this lament is a classic example of what you could call intellectual out-of-touchness. He is genuinely puzzled why so many of his fellow citizens are innumerate, can’t calculate simple odds and fall for all sorts of paranormal, astrology, snake-oil blether.

He proposes typically academic, university-level explanations for this phenomenon – such as that people find maths too cold and analytical and worry that it prevents them thinking about the big philosophical questions in life. He worries that maths has an image problem.

In other words, he fails to consider the much more obvious explanation that maths, probability and numeracy in general might be a combination of fanciful, irrelevant and deeply, deeply boring.

I use the word ‘fanciful’ deliberately. When he writes that the probability of drawing two aces in succession from a pack of cards is not (4/52 x 4/52) but (4/52 x 3/51) I do actually understand the distinction he’s making (having drawn one ace there are only 3 left and only 52 cards left) – I just couldn’t care less. I really couldn’t care less.

Or take this paragraph:

Several years ago Pete Rose set a National League record by hitting safely in forty-four consecutive games. If we assume for the sake of simplicity that he batted .300 (30 percent of the time he got a hit, 70 percent of the time he didn’t) and that he came to bat four times a game, the chances of his not getting a hit in any given game were, assuming independence, (.7)⁴ – .24… [at this point Paulos has to explain what ‘independence’ means in a baseball context: I couldn’t care less]… So the probability he would get at least one hit in any game was 1-.24 = .76. Thus, the chances of him getting a hit in any given sequence of forty-four consecutive games were (.76)⁴⁴ = .0000057, a tiny probability indeed. (p.44)

I did, in fact, understand the maths and the working out in this example. I just don’t care about the problem or the result.

For me this is a – maybe the – major flaw of this book. This is that in the blurbs on the front and back, in the introduction and all the way through the text, Paulos goes on and on about how we as a society need to be mathematically numerate because maths (and particularly probability) impinges on so many areas of our life.

But when he tries to show this – when he gets the opportunity to show us what all these areas of our lives actually are – he completely fails.

Almost all of the examples in the book are not taken from everyday life, they are remote and abstruse problems of gambling or sports statistics.

which is more likely: obtaining at least one 6 in four rolls of a single die, or obtaining at least one 12 in twenty four rolls of a pair of dice?
if 20% of drinks dispensed by a vending machine overflow their cups, what is the probability that exactly three of the next ten will overflow?
Assume there is a test for cancer which is 98% accurate i.e. if someone has cancer, the test will be positive 98 percent of the time, and if one doesn’t have it, the test will be negative 98 percent of the time. Assume further that .5 percent – one out of two hundred people – actually have cancer. Now imagine that you’ve taken the test and that your doctor sombrely informs you that you have tested positive. How depressed should you be?
What are the odds on Pete Rose getting a hit in a sequence of forty-four games?

Are these the kinds of problems you are going to encounter today? Or tomorrow? Or ever?

No. The longer the book went on, the more I realised just how little a role maths plays in my everyday life. In fact more or less the only role maths plays in my life is looking at the prices in supermarkets, where I am attracted to goods which have a temporary reduction on them. But I do that because they’re labels are coloured red, not because I calculate the savings. Being aware of the time, so I know when to do household chores or be somewhere punctually. Those are the only times I used numbers today.

5. Statistics, trade-offs and society

This feeling that the abstruseness of the examples utterly contradicts the bold claims that reading the book will help us with everyday experiences was confirmed in the final chapter, which begins with the following example.

Imagine four dice, A, B, C and D, strangely numbered as follows: A has 4 on four faces and 0 on two faces; B has 3s on all six faces; C has four faces with 2 and two faces with 6; and D has 5 on three faces and 1 on three faces…

I struggled to the end of this sentence and just thought: ‘No, no more, I don’t have to make myself feel sick and unhappy any more’ – and skipped the couple of pages detailing the fascinating and unexpected results you can get from rolling such a collection of dice.

This chapter goes on to a passage about the Prisoner’s Dilemma, a well-known problem in logic, which I have read about and instantly forgotten scores of times over the years.

Paulos gives us three or four variations on the idea, including:

Imagine you are locked up in prison by a philanthropist with 20 other people.

Or:

Imagine you are locked in a dungeon by a sadist with 20 other people.

Or:

Imagine you are one of two drug traffickers making a quick transaction on a street corner and forced to make a quick decision.

Or:

Imagine you are locked in a prison cell, and another prisoner is locked in an identical cell down the corridor.

Well, I’m not any of these things, I’m never likely to be, and I am not really interested in these fanciful speculations.

Moreover, I am well into middle age, have travelled round the world, had all sorts of jobs in companies small, large and enormous – and I am not aware of having ever been in any situation which remotely resembled any variation of the Prisoner’s Dilemma I’ve ever heard of.

In other words, to me, it is another one of the endless pile of games and puzzles which logicians and mathematicians love to spend all day playing but which have absolutely no impact whatsoever on any aspect of my life.

Pretty much all of his examples conclusively prove how remote mathematical problems and probabilistic calculation is from the everyday lives you and I lead. When he asks:

How many people would there have to be in a group in order for the probability to be half that at least two people in it have the same birthday? (p.23)

Imagine a factory which produces small batteries for toys, and assume the factory is run by a sadistic engineer… (p.117)

It dawns on me that my problem might not be that I’m innumerate, so much as I’m just uninterested in trivial or frivolous mental exercises.

Someone offers you a choice of two envelopes and tells you one has twice as much money in it as the other. (p.127)

Flip a coin continuously until a tail appears for the first time. If this doesn’t happen until the twentieth (or later) flip, you win $1 billion. If the first tail occurs before the twentieth flip, you pay $100. Would you play? (p.128)

No. I’d go and read an interesting book.

Thoughts

If Innumeracy: Mathematical Illiteracy and Its Consequences is meant to make its readers more numerate, it failed with me.

This is for a number of reasons:

crucially – because he doesn’t explain maths very well; or, the way he explained probability had lost me by about page 16 – in other words, if this is meant to be a primer for innumerate people it’s a fail
because the longer it goes on, the more convinced I became that I rarely use maths, arithmetic and probability in my day today life: whole days go by when I don’t do a single sum, and so lost all motivation to submit myself to the brain-hurting ordeal of trying to understand his examples

3. Also because the structure and presentation of the book is a mess. The book meanders through a fog of jokes, anecdotes and maths trivia, baseball stories and gossip about American politicians – before suddenly unleashing a fundamental aspect of probability theory on the unwary reader.

I’d have preferred the book to have had a clear, didactic structure, with an introduction and chapter headings explaining just what he was going to do, an explanation, say, of how he was going to take us through some basic concepts of probability one step at a time.

And then for the concepts to have been laid out very clearly and explained very clearly, from a number of angles, giving a variety of different examples until he and we were absolutely confident we’d got it – before we moved on to the next level of complexity.

The book is nothing like this. Instead it sacrifices any attempt at logical sequencing or clarity for anecdotes about Elvis Presley or UFOs, for digressions about Biblical numerology, the silliness of astrology, the long and bewildering digression about introducing a safety index for activities (summarised above), or prolonged analyses of baseball or basketball statistics. Oh, and a steady drizzle of terrible jokes.

Which two sports have face-offs?
Ice hockey and leper boxing.

Half way through the book, Paulos tells us that he struggles to write long texts (‘I have a difficult time writing at extended length about anything’, p.88), and I think it really shows.

It certainly explains why:

the blizzard of problems in coin tossing and dice rolling stopped without any warning, as he switched tone copletely, giving us first a long chapter about all the crazy irrational beliefs people hold, and then another chapter listing all the reasons why society is innumerate
the last ten pages of the book give up the attempt of trying to be a coherent narrative and disintegrate into a bunch of miscellaneous odds and ends he couldn’t find a place for in the main body of the text

Also, I found that the book was not about numeracy in the broadest sense, but mostly about probability. Again and again he reverted to examples of tossing coins and rolling dice. One enduring effect of reading this book is going to be that, the next time I read a description of someone tossing a coin or rolling a die, I’m just going to skip right over the passage, knowing that if I read it I’ll either be bored to death (if I understand it) or have an unpleasant panic attack (if I don’t).

In fact in the coda at the end of the book Paulos explicitly says it has mostly been about probability – God, I wish he’d explained that at the beginning.

Right at the very, very end he briefly lists key aspects of probability theory which he claims to have explained in the book – but he hasn’t, some of them are only briefly referred to with no explanation at all, including: statistical tests and confidence intervals, cause and correlation, conditional probability, independence, the multiplication principle, the notion of expected value and of probability distribution.

These are now names I have at least read about, but they are all concepts I am nowhere near understanding, and light years away from being able to use in practical life.

Innumeracy – or illogicality?

Also there was an odd disconnect between the broadly psychological and philosophical prose explanations of what makes people so irrational, and the incredibly narrow scope of the coin-tossing, baseball-scoring examples.

What I’m driving at is that, in the long central chapter on Pseudoscience, when he stopped to explain what makes people so credulous, so gullible, he didn’t really use any mathematical examples to disprove Freudianism or astrology or so on: he had to appeal to broad principles of psychology, such as:

people are drawn to notable exceptions, instead of considering the entire field of entities i.e.
people filter out the bad and the failed and focus on the good and the successful
people seize hold of the first available explanation, instead of considering every single possible permutation
people humanise and personalise events (‘bloody weather, bloody buses’)
people over-value coincidences

My point is that there is a fundamental conceptual confusion in the book which is revealed in the long chapter about pseudoscience which is that his complaint is not, deep down, right at bottom, that people are innumerate; it is that people are hopelessly irrational and illogical.

Now this subject – the fundamental ways in which people are irrational and illogical – is dealt with much better, at much greater length, in a much more thorough, structured and comprehensible way in Stuart Sutherland’s great book, Irrationality, which I’ll be reviewing and summarising later this week.

Innumeracy amounts to random scratches on the surface of the vast iceberg which is the deep human inability to think logically.

Conclusion

In summary, for me at any rate, this was not a good book – badly structured, meandering in direction, unable to explain even basic concepts but packed with digressions, hobby horses and cul-de-sacs, unsure of its real purpose, stopping for a long rant against pseudosciences and an even longer lament on why maths is taught so badly – it’s a weird curate’s egg of a text.

Its one positive effect was to make me want to track down and read a good book about probability.

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by Simon on March 25, 2019 • Permalink

Posted in Books, Science

Posted by Simon on March 25, 2019

https://astrofella.wordpress.com/2019/03/25/innumeracy-john-allen-paulos/

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Books & Boots

All posts tagged Mathematics

The Romantic by Hermann Broch (1931)

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Interlude: Why is the novel titled The Romantic?

The Romantic (1888) – plot summary, part two

Religion

Descriptions

Credit

20th century German literature

The Weimar Republic

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

Chapter Zero – A Head for Numbers

Chapter One – The Counter Culture

Chapter Two – Behold!

Chapter Three – Something About Nothing

Chapter Four – Life of Pi

Chapter Five – The x-factor

Chapter Six – Playtime

Chapter Seven – Secrets of Succession

Chapter Eight – Gold Finger

Chapter Nine – Chance Is A Fine Thing

Chapter Ten – Situation Normal

Chapter Eleven – The End of the Line

Thoughts

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Nature’s Numbers by Ian Stewart (1995)

1. The Natural Order

2. What Mathematics is For

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4. The Constants of Change

5. From Violins to Videos

6. Broken Symmetry

7. The Rhythm of Life

8. Do Dice Play God?

9. Drops, Dynamics and Daisies

Coda: Morphomatics

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Maths ideas from John Allen Paulos

1. Innumeracy: Mathematical Illiteracy and Its Consequences (1988)

2. A Mathematician Reads the Newspaper: Making Sense of the Numbers in the Headlines

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