Irrationality: The Enemy Within by Stuart Sutherland (1992)

The only way to substantiate a belief is to try to disprove it. (p.48)

Sutherland was 65 when he wrote this book, nearing the end of a prestigious career in psychology research. His aim was to lay out, in 23 themed chapters, all the psychological and sociological research data  from hundreds of experiments, which show just how prey the human mind is to a plethora of unconscious biases, prejudices, errors, mistakes, misinterpretations and so on – the whole panoply of ways in which the supposedly rational human beings can end up making grotesque mistakes. By the end he claims to have defined and demonstrated over 100 distinct cognitive errors humans are prone to (p.309).

I first read it in 2000 and it made a big impact on me because I didn’t really know that this entire area of study existed, and had certainly never read such a compendium of sociology and psychology experiments before.

I found the naming of the various errors particularly powerful. They reminded me of the lists of weird and wonderful Christian heresies I was familiar with from years of reading medieval history. And, after all, the two have a lot in common, both being lists of ‘errors’ which the human mind can make as it falls short of a) orthodox theology and b) optimally rational thinking, the great shibboleths of the Middle Ages and of the Modern World, respectively.

 

Reading it now, 20 years later, having brought up a couple of children and worked for a while in big government departments, I am a lot less shocked and amazed. I have witnessed at first hand the utter irrationality of small and medium-sized children – and then so many examples of the corporate conformity, avoidance of embarrassment, unwillingness to speak up, deferral to authority, and general mismanagement to be found in the civil service that, upon rereading the book, hardly any of it came as a surprise, more a confirmation of what I’ve witnessed at first hand.

But to have the errors so carefully named and defined and worked through in a structured way, with so many experiments giving such vivid proof of how useless humans are at even basic logic was still very enjoyable.

What is rationality?

You can’t define irrationality without first defining what you mean by rationality:

Rational thinking is most likely to lead to the conclusion that is correct, given the information available at the time (with the obvious rider that, as new information comes to light, you should be prepared to change your mind).

Rational action is that which is most likely to achieve your goals. But in order to achieve this, you have to have clearly defined goals. Not only that but, since most people have multiple goals, you must clearly prioritise your goals.

Few people think hard about their goals and even fewer think hard about the many possible consequences of their actions. (p.129)

Cognitive biases contrasted with logical fallacies

Before proceeding it’s important to point out that there is a wholly separate subject of logical fallacies. As part of his Philosophy A-Level my son was given a useful handout with a list of about fifty of these. But logical fallacies are not the same as cognitive biases.

A logical fallacy stems from an error in a logical argument; it is specific and easy to identify and correct. Cognitive bias derives from deep-rooted, thought-processing errors which themselves stem from problems with memory, attention, self-awareness, mental strategy and other mental mistakes. Far harder to acknowledge, in many cases, very hard to correct.

Fundamentals of irrationality

1. Innumeracy One of the largest causes of all irrational behaviour is that people by and large don’t understand statistics or maths. Thus most people are not intellectually equipped to understand the most reliable type of information available to human beings – data in the form of numbers. Instead they tend to make decisions based on a wide range of faulty and irrational psychological biases.

2. Physiology People are often influenced by physiological factors. Apart from obvious ones like tiredness or hunger, which are universally known to affect people’s cognitive abilities, there are also a) drives (direct and primal) like hunger, thirst, sex, and b) emotions (powerful but sometimes controllable) like love, jealousy, fear and – especially relevant – embarrassment: acute reluctance to acknowledge limits to your own knowledge or that you’ve made a mistake.

More seriously people can be alcoholics, drug addicts, and prey to a wide range of other obsessive behaviours, not to mention suffering from a wide range of mental illnesses or conditions which undermine any attempt at rational decision-making, such as stress, anxiety or, at the other end of the spectrum, depression and loss of interest.

3. The functional limits of consciousness Numerous experiments have shown that human beings have a limited capacity to process information. Given that people rarely have a) a sufficient understanding of the relevant statistical data, and b) the RAM capacity to process all the data required to make the optimum decision, it is no surprise that most of us fall back on all manner of more limited, non-statistical biases and prejudices when it comes to making decisions.

The wish to feel good The world is threatening, dangerous and competitive. Humans want to feel safe, secure, calm, in control. This is fair enough, but it does mean that people have a way of blocking out any kind of information which threatens them. People irrationally believe they are cleverer than they in fact are, are qualified in areas of activity of knowledge where they aren’t, people stick to bad decisions for fear of being embarrassed or humiliated, and for the same reason reject new evidence which contradicts their position.

Named types of error and bias

Jumping to conclusions Sutherland tricks the reader no page one by asking a series of questions and then pointing out, that if you tried to answer about half of them, you are a fool since they don’t contain enough information to arrive at any sort of solution. Jumping to conclusions before we have enough evidence is a basic and universal error. One way round this is to habitually use a pen and paper to set out the pros and cons of any decision, which also helps highlight areas where you realise you don’t have enough information.

The availability error All the evidence is that the conscious mind can only hold a small number of data or impressions at any one time (near the end of the book, Sutherland claims the maximum is seven items, p.319). Many errors are due to people reaching for the most available explanation, using the first thing that comes to mind, and not taking the time to investigate further and make a proper, rational survey of the information.

Many experiments show that you can unconsciously bias people by planting ideas, words or images in their minds which then directly affect decisions they take hours later about supposedly unconnected issues.

Studies show that doctors who have seen a run of a certain condition among their patients become more likely to diagnose it in patients who don’t have it. The diagnosis is more ‘available’.

The news media is hard-wired to publicise shocking and startling stories which leads to the permanent misleading of the reading public. One tourist eaten by a shark in Australia eclipses the fact that you are far more likely to die in a car crash than be eaten by a shark.

Thus ‘availability’ is also affected by impact or prominence. Experimenters read out a list of men and women to two groups without telling them that there are exactly 25 men and 25 women, and asked them to guess the ratio of the sexes. If the list included some famous men, the group was influenced to think there were more men, if the list included famous women, the group thought there are more women than men.

The entire advertising industry is based on the availability error in the way it invents straplines, catchphrases and jingles designed to pop to the front of your mind when you consider any type of product, to be – in other words – super available.

I liked the attribution of the well-known fact that retailers price goods at just under the nearest pound, to the availability error. Most of us find £5.95 much more attractive than £6. It’s because we only process the initial 5, the first digit, it is more available.

Numerous studies have shown that the effect is hugely increased under stress. Under stressful situations – in an accident – people fixate on the first solution that comes to mind and refuse to budge.

The primacy effect First impressions. Interviewers make up their minds in the first minute of an interview and then spend the rest of the time collecting data to confirm that first impression.

The anchor effect In picking a number people tend to choose one close to any number they were presented with. Two groups were asked to estimate whether the population of Turkey was a) bigger than 5 million b) less than 65 million, and what it was. The group who’d had 5 million planted in their mind hovered around 15 million, the group who’d had 65 million hovered around 35 million. They were both wrong. It is 80 million.

The halo effect People extrapolate the nature of the whole from just one quality e.g. in tests, people think attractive people must be above average in personality and intelligence although of course there is no reason why they should be. Hence this error’s alternative name, the ‘physical attractiveness stereotype’. The halo effect is fundamental to advertising which seeks to associate images of beautiful men, women, smiling children, sunlit countryside etc with the product.

The existence of the halo effect and primacy effect are both reasons why interviews are a poor way to assess candidates for jobs or places.

The devil effect Opposite of the above: extrapolating from negative appearances to the whole. This is why it’s important to dress smartly for an interview or court appearance, it really does influence. In an experiment examiners were given identical answers, but some in terrible handwriting, some in beautifully clear handwriting. Clear handwriting consistently scored higher marks despite identical factual content of the scripts.

Illusory correlation People find links between disparate phenomena which simply don’t exist, thus:

  • people exaggerate the qualities of people or things which stand out from their environments
  • people associate rare qualities with rare things

This explains a good deal of racial prejudice: a) immigrants stand out b) a handful of immigrants commit egregious behaviour – therefore it is a classic example of illusory correlation to associate the two. What is missing is taking into account all the negative examples i.e. the millions of immigrants who make no egregious behaviour and whose inclusion would give you a more accurate statistical picture. Pay attention to negative cases.

Stereotypes 1. People tend to notice anything which supports their existing opinions. 2. We notice the actions of ‘minorities’ much more than the actions of the invisible majority.

Projection People project onto neutral phenomena patterns and meanings they are familiar with or which bolster their beliefs. Compounded by –

Obstinacy Sticking to personal opinions (often made in haste / first impressions / despite all evidence to the contrary) aka The boomerang effect When someone’s opinions are challenged, they just become more obstinate about it. Aka Belief persistence. Aka pig-headedness. Exacerbated by –

Group think People associate with others like themselves, which makes them feel safe by a) confirming their beliefs and b) letting them hide in a crowd. Experiments have shown how people in self-supporting groups are liable to become more extreme in their views. Also – and I’ve seen this myself – groups will take decisions that almost everyone in the group, as individuals, know to be wrong – but no-one is prepared to risk the embarrassment or humiliation of pointing it out. The Emperor’s New Clothes. Groups are more likely to make irrational decisions than individuals are.

Confirmation bias The tendency to search for, interpret, favour, and recall information in a way that confirms one’s pre-existing beliefs or hypotheses. In an experiment people were read out a series of statements about a named person, who had a stated profession and then two adjectives describing them, one what you’d expect, the other less predictable. ‘Carol, a librarian, is attractive and serious’. When asked to do a quiz at the end of the session, participants showed a marked tendency to remember the expected adjective, and forget the unexpected one. Everyone remembered that the air stewardess was ‘attractive’ but remembered the librarian for being ‘serious’.

We remember what we expect to hear. (p.76)

Or: we remember what we remember in line with pre-existing habits of thought, values etc.

We marry people who share our opinions, we have friends with people who share our opinions, we agree with everyone in our circle on Facebook.

Self-serving biases When things go well, people take the credit, when things go badly, people blame external circumstances.

Avoiding embarrassment People obey, especially in a group situation, bad orders because they don’t want to stick out. People go along with bad decisions because they don’t want to stick out. People don’t want to admit they’ve made a mistake, in front of others, or even to themselves.

Avoiding humiliation People are reluctant to admit mistakes in front of others. And rather than make a mistake in front of others, people would rather keep quiet and say nothing (in a meeting situation) or do nothing, if everyone else is doing nothing (in an action situation). Both of these avoidances feed into –

Obedience The Milgram experiment proved that people will do any kind of atrocity for an authoritative man in a white coat. All of his students agreed to inflict life-threatening levels of electric shock on the victim, supposedly wired up in the next door room and emitting blood curdling (faked) screams of pain. 72% of Senior House Officers wouldn’t question the decision of a consultant, even if they thought he was wrong.

Conformity Everyone else is saying or doing it, so you say or do it so as not to stick out / risk ridicule.

Obedience is behaving in a way ordered by an authority figure. Conformity is behaving in a way dictated by your peers.

The wrong length lines experiment. You’re put in a room with half a dozen stooges, and shown a piece of card with a line on it and then another piece of card with three lines of different length on it, and asked which of the lines on card B is the same length as the line on card A. To your amazement, everyone else in the room chooses a line which is obviously wildly wrong. In experiments up to 75%! of people in this situation go along with the crowd and choose the line which they are sure, can see, know is wrong – because people are that easily swayed.

Sunk costs fallacy The belief that you have to continue wasting time and money on a project because you’ve invested x amount of time and money to date. Or ‘throwing good money after bad’.

Sutherland keeps cycling round the same nexus of issues, which is that people jump to conclusions – based on availability, stereotypes, the halo and anchor effects – and then refuse to change their minds, twisting existing evidence to suit them, ignoring contradictory evidence.

Misplaced consistency & distorting the evidence Nobody likes to admit (especially to themselves) that they are wrong. Nobody likes to admit (especially to themselves) that they are useless at taking decisions.

Our inability to acknowledge our own errors even to ourselves is one of the most fundamental causes of irrationality. (p.100)

And so:

  • people consistently avoid exposing themselves to evidence that might disprove their beliefs
  • on being faced with evidence that disproves their beliefs, they ignore it
  • or they twist new evidence so as to confirm their existing beliefs
  • people selectively remember their own experiences, or misremember the evidence they were using at the time, in order to validate their current decisions and beliefs
  • people will go to great lengths to protect their self-esteem

Sutherland says the best cleanser / solution / strategy to fixed and obstinate ideas is to make the time to gather as much evidence as possible and to try to disprove your own position. The best solution will be the one you have tried to demolish with all the evidence you have and still remains standing.

People tend to seek confirmation of their current hypothesis, whereas they should be trying to disconfirm it. (p.138)

Fundamental attribution error Ascribing other people’s behaviour to their character or disposition rather than to their situation. Subjects in an experiment watched two people holding an informal quiz: the first person made up questions (based on what he knew) and asked the second person who, naturally enough, hardly got any of them right. Observers consistently credited the quizzer with higher intelligence than the answerer, completely ignoring the in-built bias of the situation, and instead ascribing the difference to character.

We are quick to personalise and blame in a bid to turn others into monolithic entities which we can then define and control – this saves time and effort, and makes us feel safer and secure – whereas the evidence is that all people are capable of a wide range of behaviours depending on the context and situation.

Once you’ve pigeon-holed someone, you will tend to notice aspects of their behaviour which confirm your view – confirmation bias and/or illusory correlation and a version of the halo/devil effect. One attribute colours your view of a more complex whole.

Actor -Observer Bias Variation on the above: when we screw up we find all kinds of reasons in the situation to exonerate ourselves, we performed badly because we’re ill, jet-lagged, grandma died, reasons that are external to us. If someone else screws up, it is because they just are thick, lazy, useless. I.e. we think of ourselves as complex entities subject to multiple influences, and others as monolithic types.

False Consensus Effect Over-confidence that other people think and feel like us, that our beliefs and values are the norm – in my view one of the greatest errors of our time.

It is a variation of the ever-present Availability Error because when we stop to think about any value or belief we will tend to conjure up images of our family and friends, maybe workmates, the guys we went to college with, and so on: in other words, the people available to memory – simply ignoring the fact that these people are a drop in the ocean of the 65 million people in the UK. See Facebubble.

The False Consensus Effect reassures us that we are normal, our values are the values, we’re the normal ones: it’s everyone else who is wrong, deluded, racist, sexist, whatever we don’t approve of.

Not in Sutherland’s book, I’ve discovered some commentators naming this the Liberal fallacy:

For liberals, the correctness of their opinions – on universal health care, on Sarah Palin, on gay marriage – is self-evident. Anyone who has tried to argue the merits of such issues with liberals will surely recognize this attitude. Liberals are pleased with themselves for thinking the way they do. In their view, the way they think is the way all right-thinking people should think. Thus, “the liberal fallacy”: Liberals imagine that everyone should share their opinions, and if others do not, there is something wrong with them. On matters of books and movies, they may give an inch, but if people have contrary opinions on political and social matters, it follows that the fault is with the others. (Commentary magazine)

Self-Serving Bias People tend to give themselves credit for successes but lay the blame for failures on outside causes. If the project is a success, it was all due to my hard work and leadership. If it’s a failure, it’s due to circumstances beyond my control, other people not pulling their weight etc.

Preserving one’s self-esteem These three errors are all aspects of preserving our self-esteem. You can see why this has an important evolutionary and psychological purpose. In order to live, we must believe in ourselves, our purposes and capacities, believe our values are normal and correct, believe we make a difference, that our efforts bring results. No doubt it is a necessary belief and a collapse of confidence and self-belief can lead to depression and possibly despair. But that doesn’t make it true. People should learn the difference between having self-belief to motivate themselves, and developing the techniques to gather the full range of evidence – including the evidence against your own opinions and beliefs – which will enable them to make correct decisions.

Representative error People estimate the likelihood of an event by comparing it to an existing prototype / stereotype that already exists in our minds. Our prototype is what we think is the most relevant or typical example of a particular event or object. This often happens around notions of randomness: people have a notion of what randomness should look like i.e. utterly scrambled. But in fact plenty of random events or sequences arrange themselves into patterns we find meaningful. So we dismiss them as not really random.  I.e. we have judged them against our preconception of what random ought to look like.

Ask a selection of people which of these three sets of six coin tosses where H stands for heads, T for tails is random.

  1. TTTTTT
  2. TTTHHH
  3. THHTTH

Most people will choose 3 because it feels random. But of course all three are equally likely or unlikely.

Hindsight In numerous experiments people have been asked to predict the outcome of an event, then after the event questioned about their predictions. Most people forget their inaccurate predictions and misremember that they were accurate.

Overconfidence Most professionals have been shown to overvalue their expertise i.e. exaggerate their success rates.


Statistics

The trouble with this and Paulos’s books is that the entire area of statistics is separate and distinct from errors of thought and cognitive biases. I.e. you can imagine someone who avoids all of the cognitive and psychological errors named above, but still makes howlers when it comes to statistics simply because they’re not very good at it.

This is because the twin areas of Probability and Statistics are absolutely fraught with difficulty. Either you have been taught the correct techniques, and understand them, and practice them regularly (and both books demonstrate that even experts make howling mistakes in the handling of statistics and probability) or, like most of us, you have not.

As Sutherland points out, most people’s knowledge of statistics is non-existent. Since we live in a society whose public discourse i.e. politics, is ever more dominated by statistics…

Errors in estimating probability or misunderstanding samples, opinion polls and so on are probably a big part of irrationality, but I felt that they’re so distinct from the psychological biases discussed above, that they almost require a separate volume, or a separate ‘part’ of this volume. Briefly, common mistakes are:

  • too small a sample size
  • biased sample
  • not understanding that any combination of probabilities is less likely than either on their own, which requires an understanding of base rate or a priori probability
  • the law of large numbers – the more a probabilistic event takes place, the more likely the result will move towards the theoretical probability
  • be aware of the law of regression to the mean
  • be aware of the law of large numbers

Gambling

This is even more true of gambling. It is a highly specialised and advanced form of probability applied to games which have been pored over by very clever people for centuries. It’s not a question of a few general principles, this is a vast, book-length subject in its own right. Some points that emerge:

  • always work out the expected value of a bet i.e. the amount to be won times the probability of winning it

The two-by-two box

It’s taken me some time to understand this principle which is given in both Paulos and Sutherland.

When two elements with a yes/no result are combined, people tend to look at the most striking correlation and fixate on it. The only way to avoid the false conclusions that follow from that is to draw a 2 x 2 box and work through the figures.

Here is a table of 1,000 women who had a mammogram because their doctors thought they had symptoms of breast cancer.

Women with cancer Women with no cancer Total
Women with positive mammography 74 110 184
Women with negative mammography 6 810 816
80 920 1000

Bearing in mind that a conditional probability is saying that if X and Y are linked, then the chances of X, if Y, are so and so – i.e. the probability of X is conditional on the probability of Y – this table allows us to work out the following conditional probabilities:

1. The probability of getting a positive mammogram or test result, if you do actually have cancer, is 74 out of 80 = .92 (out of the 80 women with cancer, 74 were picked up by the test)

2. The probability of getting a negative mammogram or test result and not having cancer, is 810 out of 920 = .88

3. The probability of having cancer if you test positive, is 74 out of 184 = .40

4. The probability of having cancer if you test negative, is 6 out of 816 = .01

So 92% of women of women with cancer were picked up by the test. BUT Sutherland quotes a study which showed that a shocking 95% of doctors thought that this figure – 92% – was also the probability of a patient who tested positive having the disease. By far the majority of US doctors thought that, if you tested positive, you had a 92% chance of having cancer. They fixated on the 92% figure and transposed it from one outcome to the other, confusing the two. But this is wrong. The probability of a woman testing positive actually having cancer is given in conclusion 3 – 74 out of 184 = 40%. This is because 110 out of the total 184 women tested positive, but did not have cancer.

So if a woman tested positive for breast cancer, the chances of her actually having it are 40%, not 92%. Quite a big difference (and quite an indictment of the test, by the way). And yet 95% of doctors thought that if a woman tested positive she had a 92% likelihood of having cancer.

Sutherland goes on to quote a long list of other situations where doctors and others have comprehensively  misinterpreted the results of studies like this, with varied and sometimes very negative consequences.

The moral of the story is if you want to determine whether one event is associated with another, never attempt to keep the co-occurrence of events in your head. It’s just too complicated. Maintain a written tally of the four possible outcomes and refer to these.


Deep causes

Sutherland concludes the book by speculating that all the hundred or so types of irrationality he has documented can be attributed to five fundamental causes:

  1. Evolution We evolved to make snap decisions, we are brilliant at processing visual information and responding before we’re even aware of it. Conscious thought is slower, and the conscious application of statistics, probability, regression analysis and so on is very slow and laborious. Most people never acquire it.
  2. Brain structure As soon as we start perceiving, learning and remembering the world around us brain cells make connections. The more the experience is repeated, the stronger the connections become. Routines and ruts form, which are hard to budge.
  3. Heuristics Everyone develops mental short-cuts, techniques to help make quick decisions. Not many people bother with the laborious statistical techniques for assessing relative benefits which Sutherland describes.
  4. Failure to use elementary probability and elementary statistics Ignorance is another way of describing this, mass ignorance. Sutherland (being an academic) blames the education system. I, being a pessimist, attribute it to basic human nature. Lots of people just are lazy, lots of people just are stupid, lots of people just are incurious.
  5. Self-serving bias In countless ways people are self-centred, overvalue their judgement and intelligence, overvalue the beliefs of their in-group, refuse to accept it when they’re wrong, refuse to make a fool of themselves in front of others by confessing error or pointing out errors in others (especially the boss) and so on.

I would add two more:

Suggestibility. Humans are just tremendously suggestible.

Say a bunch of positive words to test subjects, then ask them questions on an unrelated topic: they’ll answer positively. Take a different representative sample of subjects and run a bunch of negative words past them, then ask them the same unrelated questions, and their answers will be measurably more negative.

Ask subjects how they get a party started and they will talk and behave extrovert to the questioner. Ask them how they cope with feeling shy and ill at ease at parties, and they will tend to act shy and speak quieter. The initial terms or anchor defines the ensuing conversation.

In one experiment a set of subjects were shown one photo of a car crash. Half were asked to describe what they think happened when one car hit another; the other half were asked to describe what they thought happened when one car smashed into the other. The ones given the word ‘smashed’ gave much more melodramatic accounts. Followed up a week later, the subjects were asked to describe what they remembered of the photo. The subjects given the word ‘hit’ fairly accurately described it, whereas the subjects given the word ‘smashed’ invented all kinds of details like a sea of broken glass around the vehicles which simply wasn’t there, which their imaginations had invented, all at the prompting of one word.

Many of the experiments Sutherland quotes demonstrate what you might call higher-level biases: but underling many of them is this simple-or-garden observation, that people are tremendously easily swayed, by both external and internal causes, away from the line of cold logic.

Anthropomorphism Another big underlying cause is anthropomorphism, namely the attribution of human characteristics to objects, events, chances, odds and so on. In other words, people really struggle to accept the high incidence of random accidents. Almost everyone attributes a purpose or intention to almost everything that happens. This means our perceptions of almost everything in life are skewed from the start.

During the war Londoners devised innumerable theories about the pattern of German bombing. After the war, when Luftwaffe records were analysed, it showed the bombing was more or less at random.

The human desire to make sense of things – to see patterns where none exists or to concoct theories… can lead people badly astray. (p.267)

Suspending judgement is about the last thing people are capable of. People are extremely uneasy if things are left unexplained. Most people rush to judgement like water into a sinking ship.

Cures

  • keep an open mind
  • reach a conclusion only after reviewing all the possible evidence
  • it is a sign of strength to change one’s mind
  • seek out evidence which disproves your beliefs
  • do not ignore or distort evidence which disproves your beliefs
  • never make decisions in a hurry or under stress
  • where the evidence points to no obvious decision, don’t take one
  • learn basic statistics and probability
  • substitute mathematical methods (cost-benefit analysis, regression analysis, utility theory) for intuition and subjective judgement

Comments on the book

Out of date

Irrationality was first published in 1992 and this makes the book dated in several ways (maybe this is why the first paperback edition was published by upmarket mass publisher Penguin, whereas the most recent edition was published by the considerably more niche publisher, Pinter & Martin).

In the chapter about irrational business behaviour he quotes quite a few examples from the 1970s and the oil crisis of 1974. These and other examples – such as the long passage about how inefficient the civil service was in the early 1970s – feel incredibly dated now.

And the whole thing was conceived, researched and written before there was an internet or any of the digital technology we take for granted nowadays. Can’t help make wonder how the digital age has changed or added to the long list of biases, prejudices and faulty thinking he gives, and what errors of reason have emerged specific to our fabulous digital technology.

Grumpy

But it also has passages where Sutherland extrapolates out to draw general conclusions and some of these sound more like the grumblings of a grumpy old man than anything based on evidence.

Thus Sutherland whole-heartedly disapproves of ‘American’ health fads, dismisses health foods as masochistic fashion and is particularly scathing about jogging. He thinks ‘fashion’ in any sphere of life is ludicrously irrational. He is dismissive of doctors who he accuses of rejecting statistical evidence, refusing to share information with patients and wildly over-estimating their own diagnostic abilities.

He thinks the publishers of learned scientific journals are more interested in making money out of scientists than in ‘forwarding the progress of science’ (p.185). He thinks the higher average pay that university graduates tend to get is unrelated to their attendance at university and more to do with having well connected middle and upper middle class parents, and thus considers the efforts of successive Education Secretaries to introduce student loans to be unscientific and innumerate (p.186). He criticises Which consumer magazine for using too small samples in its testing (p.215). In an extended passage he summarises Leslie Chapman’s blistering (and very out of date) critique of the civil service, Your Disobedient Servant published in 1978 (pp.69-75).

He really has it in for psychoanalysis which he accuses of all sorts of irrational thinking such as projecting, false association, refusal to investigate negative instances, failing to take into account the likelihood that the patient would have improved anyway, and so on. Half way through the book he gives a thumbnail summary:

Self-deceit exists on a massive scale: Freud was right about that. Where he went wrong was in attributing it all to the libido, the underlying sex drive. (p.197)

In other words, the book is liberally sprinkled with Sutherland’s own personal opinions, which sometimes risk giving it a crankish feel.

On the other hand it’s surprising to see how some hot button issues haven’t changed at all. In the passage about the Prisoners’ Dilemma, Sutherland takes as a real life example the problem the nations of the world were having in 1992 in agreeing to cut back carbon dioxide emissions. Sound familiar?

He also states that the single biggest factor undermining international co-operation was America’s refusal to sign global treaties to limit global warming. In 1992! Plus ça change.

Against stupidity the gods themselves contend in vain

And finally, these are the mistakes made by the most intelligent and best educated among us, people trained to assess and act on evidence.

Neither this nor John Allen Paulos’s books take into account the obvious fact that lots of people are stupid. They begin with poor genetic material, are raised in families where no-one cares about education, are let down by poor schools, and are excluded or otherwise demotivated, with the result that :

  • the average reading age in the UK is 9
  • about one in five Britons (over ten million) are functionally illiterate, and probably about the same rate innumerate

which all adds to the general festival of idiocy.

Trying to keep those pesky cognitive errors at bay (in fact The Witch by Pieter Bruegel the Elder)

Trying to keep those pesky cognitive errors at bay (otherwise known as The Witch by Pieter Bruegel the Elder)


Related link

Reviews of other science books

Cosmology

Environment / human impact

Genetics

  • The Double Helix by James Watson (1968)

Maths

Particle physics

Psychology

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

Alexander Bellos (born in 1969) is a British writer and broadcaster. He is the author of books about Brazil and mathematics, as well as having a column in The Guardian newspaper. After adventures in Brazil (see his Wikipedia page) he returned to England in 2007 and wrote this, his first book. It spent four months in the Sunday Times bestseller list and led on to five more popular maths books.

It’s a hugely enjoyable read for three reasons:

  1. Bellos immediately establishes a candid, open, good bloke persona, sharing stories from his early job as a reporter on the Brighton Argus, telling some colourful anecdotes about his time in Brazil and then being surprisingly open about the way that, when he moved back to Britain, he had no idea what to do. The tone of the book is immediately modern, accessible and friendly.
  2. However this doesn’t mean he is verbose. The opposite. The book is packed with fascinating information. Every single paragraph, almost every sentence contains a fact or insight which makes you sit up and marvel. It is stufffed with good things.
  3. Lastly, although its central theme is mathematics, it approaches this through a wealth of information from the humanities. There is as much history and psychology and anthropology and cultural studies and philosophy as there is actual maths, and these are all subjects which the average humanities graduate can immediately relate to and assimilate.

Chapter Zero – A Head for Numbers

Alex meets Pierre Pica, a linguist who’s studied the Munduruku people of the Amazon and discovered they have little or no sense of numbers. They only have names for numbers up to five. Also, they cluster numbers together logarithmically i.e. the higher the number, the closer together they clustered them. Same thing is done by kindergarten children who only slowly learn that numbers are evenly spaced, in a linear way.

This may be because small children and the Munduruku don’t count so much as estimate using the ratios between numbers.

It may also be because above a certain number (five) Stone Age man needed to make quick estimates along the lines of, Are there more wild animals / members of the other gang, than us?

Another possibility is that distance appears to us to be logarithmic due to perspective: the first fifty yards we see in close detail, the next fifty yards not so detailed, beyond 100 yards looking smaller, and so on.

It appears that we have to be actively taught when young to overcome our logarithmic instincts, and to apply the rule that each successive whole number is an equal distance from its predecessor and successor i.e. the rational numbers lies along a straight line at regular intervals.

More proof that the logarithmic approach is the deep, hard-wired one is the way most of us revert to its perspective when considering big numbers. As John Allen Paulos laments, people make no end of fuss about discrepancies between 2 or 3 or 4 – but are often merrily oblivious to the difference between a million or a billion, let alone a trillion. For most of us these numbers are just ‘big’.

He goes on to describe experiments done on chimpanzees, monkeys and lions which appear to show that animals have the ability to estimate numbers. And then onto experiments with small babies which appear to show that as soon as they can focus on the outside world, babies can detect changes in number of objects.

And it appears that we also have a further number skill, that guesstimating things – the journey takes 30 or 40 minutes, there were twenty or thirty people at the party, you get a hundred, maybe hundred and fifty peas in a sack. When it comes to these figures almost all of us give rough estimates.

To summarise:

  • we are sensitive to small numbers, acutely so of 1, 2, 3, 4, less so of 5, 6, 7, 8, 9
  • left to our own devices we think logarithmically about larger numbers i.e lose the sense of distinction between them, clump them together
  • we have a good ability to guesstimate medium size numbers – 30, 40, 100

But it was only with the invention of notation, a way of writing numbers down, that we were able to create the linear system of counting (where every number is 1 larger than its predecessor, laid out in a straight line, at regular intervals).

And that this cultural invention enabled human beings to transcend our vague guesstimating abilities, and laid the basis for the systematic manipulation of the world which followed

Chapter One – The Counter Culture

The probable origins of counting lie in stock taking in the early agricultural revolution some 8,000 years ago.

We nowadays count using a number base 10 i.e. the decimal system. But other bases have their virtues, especially base 12. It has more factors i.e. is easier to divide: 12 can be divided neatly by 2, 3, 4 and 6. A quarter of 10 is 2.5 but of 12 is 3. A third of 10 is 3.333 but of 12 is 4. Striking that a version of the duodecimal system (pounds, shillings and pence) hung on in Britain till we finally went metric in the 1970s. There is even a Duodecimal Society of America which still actively campaigns for the superiority of a base 12 counting scheme.

Bellos describes a bewildering variety of other counting systems and bases. In 1716 King Charles XII of Sweden asked Emmanuel Swedenborg to devise a new counting system with a base of 64. The Arara in the Amazon count in pairs, the Renaissance author Luca Paccioli was just one of hundreds who have devised finger-based systems of counting – indeed, the widespread use of base 10 probably stems from the fact that we have ten fingers and toes.

He describes a complicated Chinese system where every part of the hand and fingers has a value which allows you to count up to nearly a billion – on one hand!

The Yupno system which attributes a different value for parts of the body up to its highest number, 33, represented by the penis.

Diagram showing numbers attributed to parts of the body by the Yupno tribe

Diagram showing numbers attributed to parts of the body by the Yupno tribe

There’s another point to make about his whole approach which comes out if we compare him with the popular maths books by John Allen Paulos which I’ve just read.

Paulos clearly sees the need to leaven his explanations of comparative probability and Arrow’s Theorem and so on with lighter material and so his strategy is to chuck into his text things which interest him: corny jokes, anecdotes about baseball, casual random digressions which occur to him in mid-flow. But al his examples clearly 1. emanate from Paulos’s own interests and hobby horses (especially baseball) and 2. they are tacked onto the subjects being discussed.

Bellos, also, has grasped that the general reader needs to be spoonfed maths via generous helpings of other, more easily digestible material. But Bellos’s choice of material arises naturally from the topic under discussion. The humour emerges naturally and easily from the subject matter instead of being tacked on in the form of bad jokes.

You feel yourself in the hands of a master storyteller who has all sorts of wonderful things to explain to you.

In fourth millennium BC, an early counting system was created by pressing a reed into soft clay. By 2700 BC the Sumerians were using cuneiform. And they had number symbols for 1, 10, 60 and 3,600 – a mix of decimal and sexagesimal systems.

Why the Sumerians grouped their numbers in 60s has been described as one of the greatest unresolved mysteries in the history of arithmetic. (p.58)

Measuring in 60s was inherited by the Babylonians, the Egyptians and the Greeks and is why we still measure hours in 60 minutes and the divisions of a circle by 360 degrees.

I didn’t know that after the French Revolution, when the National Convention introduced the decimal system of weights and measures, it also tried to decimalise time, introducing a new system whereby every day would be divided into ten hours, each of a hundred minutes, each divided into 100 seconds. Thus there were a very neat 10 x 100 x 100 = 100,000 seconds in a day. But it failed. An hour of 60 minutes turns out to be a deeply useful division of time, intuitively measurable, and a reasonable amount of time to spend on tasks. The reform was quietly dropped after six months, although revolutionary decimal clocks still exist.

Studies consistently show that Chinese children find it easier to count than European children. This may be because of our system of notation, or the structure of number names. Instead of eleven or twelve, Chinese, Japanese and Koreans say the equivalent of ten one, ten two. 21 and 22 become two ten one and two ten two. It has been shown that this makes it a lot simpler and more intuitive to do basic addition and subtraction.

Bellos goes on to describe the various systems of abacuses which have developed in different cultures, before explaining the phenomenal popularity of abacus counting, abacus clubs, and abacus championships in Japan which helps kids develop the ability to perform anzan, using the mental image of an abacus to help its practitioners to sums at phenomenal speed.

Chapter Two – Behold!

The mystical sense of the deep meaning of numbers, from Pythagoras with his vegetarian religious cult of numbers in 4th century BC Athens to Jerome Carter who advises leading rap stars about the numerological significance of their names.

Euclid and the elegant and pure way he deduced mathematical theorems from a handful of basic axioms.

A description of the basic Platonic shapes leads into the nature of tessalating tiles, and the Arab pioneering of abstract design. The complex designs of the Sierpinski carpet and the Menger sponge. And then the complex and sophisticated world of origami, which has its traditionalists, its pioneers and surprising applications to various fields of advanced science, introducing us to the American guru of modern origami, Robert Lang, and the Japanese rebel, Kazuo Haga, father of Haga’s Theorem.

Chapter Three – Something About Nothing

A bombardment of information about the counting systems of ancient Hindus, Buddhists, about number symbols in Sanskrit, Hebrew, Greek and Latin. How the concept of zero was slowly evolved in India and moved to the Muslim world with the result that the symbols we use nowadays are known as the Arabic numerals.

A digression into ‘a set of arithmetical tricks known as Vedic Mathematics ‘ devised by a young Indian swami at the start of the twentieth century, Bharati Krishna Tirthaji, based on a series of 16 aphorisms which he found in the ancient holy texts known as the Vedas.

Shankaracharya is a commonly used title of heads of monasteries called mathas in the Advaita Vedanta tradition. Tirthaji was the Shankaracharya of the monastery at Puri. Bellos goes to visit the current Shankaracharya who explains the closeness, in fact the identity, of mathematics and Hindu spirituality.

Chapter Four – Life of Pi

An entire chapter about pi which turns out not only to be a fundamental aspect of calculating radiuses and diameters and volumes of circles and cubes, but also to have a long history of mathematicians vying with each other to work out its value to as many decimal places as possible (we currently know the value of pi to 2.7 trillion decimal places) and the surprising history of people who have set records reciting the value if pi.

Thus, in 2006, retired Japanese engineer Akira Haraguchi set a world record for reciting the value of pi to the first 100,000 decimal places from memory! It took 16 hours with five minute beaks every two hours to eat rice balls and drink some water.

There are several types or classes of numbers:

  • natural numbers – 1, 2, 3, 4, 5, 6, 7…
  • integers – all the natural numbers, but including the negative ones as well – …-3, -2, -1, 0, 1, 2, 3…
  • fractions
  • which are also called rational numbers
  • numbers which cannot be written as fractions are called irrational numbers
  • transcendent numbers – ‘a transcendental number is an irrational number that cannot be described by an equation with a finite number of terms’

The qualities of the heptagonal 50p coin and the related qualities of the Reuleux triangle.

Chapter Five – The x-factor

The origin of algebra (in Arab mathematicians).

Bellos makes the big historical point that for the Greeks (Pythagoras, Plato, Euclid) maths was geometric. They thought of maths as being about shapes – circles, triangles, squares and so on. These shapes had hidden properties which maths revealed, thus giving – the Pythagoreans thought – insight into the secret deeper values of the world.

It is only with the introduction of algebra in the 17th century (Bellos attributes its widespread adoption to Descartes’s Method in the 1640s) that it is possible to fly free of shapes into whole new worlds of abstract numbers and formulae.

Logarithms turn the difficult operation of multiplication into the simpler operation of addition. If X x Y = Z, then log X + log Y = log Z. They were invented by a Scottish laird John Napier, and publicised in a huge book of logarithmic tables published in 1614. Englishman Henry Briggs established logarithms to base 10 in 1628. In 1620 Englishman Edmund Gunter marked logarithms on a ruler. Later in the 1620s Englishman William Oughtred placed two logarithmic rulers next to each other to create the slide rule.

Three hundred years of dominance by the slide rule was brought to a screeching halt by the launch of the first pocket calculator in 1972.

Quadratic equations are equations with an x and an x², e.g. 3x² + 2x – 4 = 0. ‘Quadratics have become so crucial to the understanding of the world, that it is no exaggeration to say that they underpin modern science’ (p.200).

Chapter Six – Playtime

Number games. The origin of Sudoku, which is Japanese for ‘the number must appear only once’. There are some 5 billion ways for numbers to be arranged in a table of nine cells so that the sum of any row or column is the same.

There have, apparently, only been four international puzzle crazes with a mathematical slant – the tangram, the Fifteen puzzle, Rubik’s cube and Sudoku – and Bellos describes the origin and nature and solutions to all four. More than 300 million cubes have seen sold since Ernö Rubik came up with the idea in 1974. Bellos gives us the latest records set in the hyper-competitive sport of speedcubing: the current record of restoring a copletely scrambled cube to order (i.e. all the faces of one colour) is 7.08 seconds, a record held by Erik Akkersdijk, a 19-year-old Dutch student.

A visit to the annual Gathering for Gardner, honouring Martin Gardner, one of the greatest popularisers of mathematical games and puzzles who Bellos visits. The origin of the ambigram, and the computer game Tetris.

Chapter Seven – Secrets of Succession

The joy of sequences. Prime numbers.

The fundamental theorem of arithmetic – In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers.

The Goldbach conjecture – one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that, Every even integer greater than 2 can be expressed as the sum of two primes. The conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort.

Neil Sloane’s idea of persistence – The number of steps it takes to get to a single digit by multiplying all the digits of the preceding number to obtain a second number, then multiplying all the digits of that number to get a third number, and so on until you get down to a single digit. 88 has a persistence of three.

88 → 8 x 8 = 64 → 6 x 4 = 24 → 2 x 4 = 8

John Horton Conway’s idea of the powertrain – For any number abcd its powertrain goes to abcd, in the case of numbers with an odd number of digits the final one has no power, abcde’s powertrain is abcde.

The Recamán sequence Subtract if you can, unless a) it would result in a negative number or b) the number is already in the sequence. The result is:

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11….

Gijswijt’s sequence a self-describing sequence where each term counts the maximum number of repeated blocks of numbers in the sequence immediately preceding that term.

1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, …

Perfect number A perfect number is any number that is equal to the sum of its factors. Thus 6 – its factors (the numbers which divided into it) are 1, 2 and 3. Which also add up to (are the sum of) 6. The next perfect number is 28 because its factors – 1, 2, 4, 7, 14 – add up to 28. And so on.

Amicable numbers A number is amicable if the sum of the factors of the first number equals the second number, and if the sum of the factors of the second number equals the first. The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Added together these make 284. The factors of 284 are 1, 2, 4, 71 and 142. Added together they make 220!

Sociable numbers In 1918 Paul Poulet invented the term sociable numbers. ‘The members of aliquot cycles of length greater than 2 are often called sociable numbers. The smallest two such cycles have length 5 and 28’

Mersenne’s prime A prime number which can be written in the form 2n – 1 a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, … and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, …

These and every other sequence ever created by humankind are documented on The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane’s. This is an online database of integer sequences, created and maintained by Neil Sloane while a researcher at AT&T Labs.

Chapter Eight – Gold Finger

The golden section a number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.

Phi The number is often symbolized using phi, after the 21st letter of the Greek alphabet. In an equation form:

a/b = (a+b)/a = 1.6180339887498948420 …

As with pi (the ratio of the circumference of a circle to its diameter), the digits go on and on, theoretically into infinity. Phi is usually rounded off to 1.618.

The Fibonnaci sequence Each number in the sequence is the sum of the two numbers that precede it. So the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it is Xn+2= Xn+1 + Xn.

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

Chapter Nine – Chance Is A Fine Thing

A chapter about probability and gambling.

Impossibility has a value 0, certainty a value 1, everything else is in between. Probabilities can be expressed as fractions e.g. 1/6 chance of rolling a 6 on a die, or as percentages, 16.6%, or as decimals, 0.16…

The probability is something not happening is 1 minus the probability of that thing happening.

Probability was defined and given mathematical form in 17th century. One contribution was the questions the Chevalier de Méré asked the mathematical prodigy Blaise Pascal. Pascal corresponded with his friend, Pierre de Fermat, and they worked out the bases of probability theory.

Expected value is what you can expect to get out of a bet. Bellos takes us on a tour of the usual suspects – rolling dice, tossing coins, and roulette (invented in France).

Payback percentage if you bet £10 at craps, you can expect – over time – to receive an average of about £9.86 back. In other words craps has a payback percentage of 98.6 percent. European roulette has a payback percentage of 97.3 percent. American roulette, 94.7 percent. On other words, gambling is a fancy way of giving your money away. A miserly slot machine has a payback percentage of 85%. The National Lottery has a payback percentage of 50%.

The law of large numbers The more you play a game of chance, the more likely the results will approach the statistical probability. Toss a coin three times, you might get three heads. Toss a coin a thousand times, the chances are you will get very close the statistical probability of 50% heads.

The law of very large numbers With a large enough sample, outrageous coincidences become likely.

The gambler’s fallacy The mistaken belief that, if something happens more frequently than normal during a given period, it will happen less frequently in the future (or vice versa). In other words, that a random process becomes less random, and more predictable, the more it is repeated.

The birthday paradox The probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. (These conclusions are based on the assumption that each day of the year (excluding February 29) is equally probable for a birthday.) In other words you only need a group of 23 people to have an evens chance that two of them share a birthday.

The drunkard’s walk

The difficulty of attaining true randomness and the human addiction to finding meaning in anything.

The distinction between playing strategy (best strategy to win a game) and betting strategy (best strategy to maximise your winnings), not always the same.

Chapter Ten – Situation Normal

Carl Friedrich Gauss, the bell curve, normal distribution aka Gaussian distribution. Normal or Gaurrian distribution results in a bell curve. Bellos describes the invention and refinement of the bell curve (he explains that ‘the long tail’ results from a mathematician who envisioned a thin bell curve as looking like two kangaroos facing each other with their long tails heading off in opposite directions). And why

Regression to the mean – if the outcome of an event is determined at least in part by random factors, then an extreme event will probably be followed by one that is less extreme. And recent devastating analyses which show how startlingly random sports achievements are, from leading baseball hitters to Simon Kuper and Stefan Szymanski’s analysis of the form of the England soccer team.

Chapter Eleven – The End of the Line

Two breakthroughs which paved the way for modern i.e. 20th century, maths: the invention of non-Euclidean geometry, specifically the concept of hyperbolic geometry. To picture this draw a triangle on a Pringle. it is recognisably a triangle but all its angles do not add up to 180°, therefore it defies, escapes, eludes all the rule of Euclidean geometry, which were designed for flat 2D surfaces.

Bellos introduces us to Daina Taimina, a maths prof at Cornell University, who invented a way of crocheting hyperbolic surfaces. The result looks curly, like curly kale or the surface of coral.

Anyway, the breakaway from flat 2-D Euclidean space led to theories about curved geometry, either convex like a sphere, or hyperbolic like the pringle. It was this notion of curved space, which paved the way for Einstein’s breakthrough ideas in the early 20th century.

The second big breakthrough was Georg Cantor’s discovery that you can have many different types of infinity. Until Cantor the mathematical tradition from the ancient Greeks to Galileo and Newton had fought shy of infinity which threatened to disrupt so many formulae.

Cantor’s breakthrough was to stop thinking about numbers, and instead think of sets. This is demonstrated through the paradoxes of Hilbert’s Hotel. You need to buckle your safety belt to understand it.

Thoughts

This is easily the best book about maths I’ve ever read. It gives you a panoramic history of the subject which starts with innumerate cavemen and takes us to the edge of Einstein’s great discoveries. But Bellos adds to it all kinds of levels and abilities.

He is engaging and candid and funny. He is fantastically authoritative, taking us gently into forests of daunting mathematical theory without placing a foot wrong. He’s a great explainer. He knows a good story when he sees one, and how to tell it engagingly. And in every chapter there is a ‘human angle’ as he describes his own personal meetings and interviews with many of the (living) key players in the world of contemporary maths, games and puzzles.

Like the Ian Stewart book but on a vastly bigger scale, Bellos makes you feel what it is like to be a mathematician, not just interested in nature’s patterns (the basis of Stewart’s book, Nature’s Numbers) but in the beauty of mathematical theories and discoveries for their own sakes. (This comes over very strongly in chapter seven with its description of some of the weirdest and wackiest number sequences dreamed up by the human mind.) I’ve often read scientists describing the beauty of mathematical theories, but Bellos’s book really helps you develop a feel for this kind of beauty.

For me, I think three broad conclusions emerged:

1. Most mathematicians are in it for the fun. Setting yourself, and solving, mathematical puzzles is obviously extremely rewarding. Maths includes the vast territory of puzzles and games, such as the Sudoku and so on he describes in chapter six. Obviously it has all sorts of real-world application in physics, engineering and so on, but Bellos’s book really brings over that a true understanding of maths begins in puzzles, games and patterns, and often remains there for a lifetime. Like everything else maths is no highly professionalised the property of tenured professors in universities; and yet even to this day – as throughout its history – contributions can be made by enthusiastic amateurs.

2. As he points out repeatedly, many insights which started out as the hobby horses of obsessives, or arcane breakthroughs on the borders of our understanding, and which have been airily dismissed by the professionals, often end up being useful, having applications no-one dreamed of. Either they help unravel aspects of the physical universe undreamed of when they were discovered, or have been useful to human artificers. Thus the development of random number sequences seemed utterly pointless in the 19th century, but now underlies much internet security.

On a profounder note, Bellos expresses the eerie, mystical sense many mathematicians have that it seems so strange, so pregnant with meaning, that so many of these arcane numbers end up explaining aspects of the world their inventors knew nothing of. Ian Stewart has an admirably pragmatic explanation for this: he speculates that nature uses everything it can find in order to build efficient life forms. Or, to be less teleological, over the past 3 and a half billion years, every combination of useful patterns has been tried out. Given this length of time, and the incalculable variety of life forms which have evolved on this planet, it would be strange if every number system conceivable by one of those life forms – humankind – had not been tried out at one time or another.

3. My third conclusion is that, despite John Allen Paulos’s and Bellos’s insistence, I do not live in a world ever-more bombarded by maths. I don’t gamble on anything, and I don’t follow sports – the two biggest popular areas where maths is important – and the third is the twin areas of surveys and opinion polls (55% of Americans believe in alien abductions etc etc) and the daily blizzard of reports (for example, I see in today’s paper that the ‘Number of primary school children at referral units soars’).

I register their existence but they don’t impact on me for the simple reason that I don’t believe any of them. In 1992 every opinion poll said John Major would lose the general election, but he won with a thumping majority. Since then I haven’t believed any poll about anything. For example almost all the opinion polls predicted a win for Remain in the Brexit vote. Why does any sane person believe opinion polls?

And ‘new and shocking’ reports come out at the rate of a dozen a day and, on closer examination, lots of them turn out to be recycled information, or much much more mundane releases of data sets from which journalists are paid to draw the most shocking and extreme conclusions. Some may be of fleeting interest but once you really grasp that the people reporting them to you are paid to exaggerate and horrify, you soon learn to ignore them.

If you reject or ignore these areas – sport, gambling and the news (made up of rehashed opinion polls, surveys and reports) – then unless you’re in a profession which actively requires the sophisticated manipulation of figures, I’d speculate that most of the rest of us barely come into contact with numbers from one day to the next.

I think that’s the answer to Paulos and Bellos when they are in ‘why aren’t more people mathematically numerate?’ mode – 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 it may be to the professionals – maths is more or less irrelevant to most of our day to day lives, most of the time.


Related links

Reviews of other science books

Cosmology

Environment / human impact

Genetics

  • The Double Helix by James Watson (1968)

Maths

Particle physics

Psychology

  • Irrationality: The Enemy Within by Stuart Sutherland (1992)

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:

  1. Examples and principles
  2. Probability and coincidence
  3. Pseudoscience
  4. Whence innumeracy
  5. 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)is 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)= .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)x (.8)7. But there are many different ways for exactly three of the ten cups to overflow, each way having probability (.2)x (.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)x (.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.

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

  1. working out the risk of an activity (i.e. number of deaths or accidents per person doing the activity)
  2. converting that into a logarithmic value (just to make sure than nobody understands it) and then
  3. 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:

  1. 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
  2. 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.


Related links

Reviews of other science books

Cosmology

Environment / human impact

Genetics

  • The Double Helix by James Watson (1968)

Maths

Particle physics

Psychology

  • Irrationality: The Enemy Within by Stuart Sutherland (1992)

Casino Royale by Ian Fleming (1953)

‘It’s not difficult to get a Double O number if you’re prepared to kill people.’ (p.64)

The casino in question is in the fictional French town of Royale-les-Eaux, just north of Dieppe, near the mouth of the river Somme (p.34), based on the holiday resorts of Deaville and Trouville – ie not the sunny south of France

James Bond is an agent for the British Secret Service. Their offices are in a gloomy building overlooking Regents Park. Its head is ‘M’ (p.14) whose personal secretary is Miss Moneypenny (p.23) who ‘would have been desirable but for eyes which were cool and direct and quizzical’. Bond has a Double O number because he has killed in the line of duty: to be precise, a Japanese cipher expert in New York and a Norwegian double agent in Stockholm (p.64 and p. 142).

Bond Biography

Bond lives in a flat in Chelsea. His only personal hobby is ‘one of the last of the 4.5 litre Bentleys with the supercharger by Amherst Villiers’, a battleship convertible coupé which he bought in 1933 (p.36). He is given penetrating awareness of everything around him, especially other people’s appearances, particularly women (eg the page-long description of Vesper Lynd pp.38-39).

Vesper thinks Bond looks like a cold, ruthless version of the popular singer and pianist Hoagy Carmichael (p.40), though when Bond himself looks in the mirror he sees cold grey-blue eyes, and a vertical scar down his right cheek, not much like Carmichael (p.57). Asleep, when the warmth and humour have left his eyes, Bond’s features relapse ‘into a taciturn mask, ironical, brutal, and cold.’ (p.13) He is a ‘harsh, cold’ man (p.151).

Le Chiffre

Bond has been sent to Royale-les-Eaux on a mission. The man known as ‘Le Chiffre’ has risen from being a Displaced Person after the War, to become one of the KGB’s top agents in France and undercover paymaster to the 50,000-strong communist-controlled Trade Union of Workers of Alsace, on the border with Germany and therefore an important fifth column if war with Russia breaks out. He is controlled by KGB ‘Leningrad Section III’.

Le Chiffre is a clever man, a cunning strategist, a cool gambler. But he has made a bad mistake. He embezzled a big sum of funds from the Union – funds ultimately belonging to the Russians. With them he bought a chain of a dozen or so brothels and porn shops. Unfortunately, soon afterwards, the French government passed a law banning both brothels and porn. He lost the lot. In fear of what will happen when his Soviet paymasters find out, le Chiffre travels to Royale-les-Eaux (which has become a notorious high-stakes gambling centre) and, in the time-honoured fashion of embezzlers who need to pay back their funds, is hoping to get lucky in the casino and win back the money.

The mission

Bond’s mission is to beat le Chiffre at the gambling tables. To humiliate a major Soviet agent and the large communist union he manages, probably leading to le Chiffre being eliminated by his own side, an organisational and propaganda victory for our side. The execution would be done by SMERSH, the Soviet execution agency – a word formed by joining two Russian words smyert shpionam Death to spies! (history & overview given on pages 21 and 147).

Bond has been handed a large amount to gamble with and the ‘cover’ of being a playboy millionaire inheritor of a large fortune in Jamaica. He is helped by Mathis, an agent from the French Deuxième Bureau, and Felix Leiter, from the American CIA (full description page 53).

A connoisseur spy

‘I take a ridiculous pleasure in what I eat and drink.’ (p.61)

Bond is a connoisseur of good food and drink, of guns, cars and women. Rereading Casino Royale it struck me that:

  • In previous literature, this level of connoisseurship was restricted to aristocratic characters, and not a usual characteristic of the special agent genre, as embodied by John Buchan, Bulldog Drummond or the ordinary bloke heroes of Eric Ambler. One element of Bond’s success is combining the visceral excitement of the spy thriller – traditionally thought of as a pulpy or low genre – with a level of upper-class connoisseurship previously restricted to more high-brow literature.
  • Bond really enjoys the things he likes, and Fleming manages to convey this enjoyment very powerfully. There are not that many stereotypical thriller scenes (one bomb goes off, there’s a very long card game, a car chase and an extended torture scene). What dominates the text is Bond’s supremely sensual enjoyment of what he likes: food, cigarettes, fast cars, fancy drinks, looking at a beautiful woman in expensive clothes.

Bond’s likes

In fact, the word ‘like’ crops up frequently. ‘Bond liked to make a good breakfast’ (p.28): in this instance, half a pint of iced orange juice, three scrambled eggs and bacon, and a double portion of coffee without sugar, followed by the first cigarette of the day, ‘a Balkan and Turkish mixture made for him by Morlands of Grosvenor Street’.

The pleasure of good food

When he orders paté de foie gras he makes sure it comes with a porcelain pot of very hot water to dip the knife in so it will cut through the paté more easily (p.45). He gives the barman at the casino very precise instructions to make him a cocktail from – three measures of Gordon’s, one of vodka, half a measure of Kina Lillet, shaken very well until it’s ice-cold, then garnished with a large thin slice of lemon-peel. ‘I never have more than one drink before dinner. But I do like that one to be large and very strong and very cold and very well-made.’ (p.51)

He has one large dinner with the woman assigned to assist him on the mission, Vesper Lynd, which is described in loving detail: Vesper orders caviar (Bond ensures it comes with plenty of toast, always the issue with caviar) then plain grilled rognon de veau with pommes soufflés, and for dessert fraises de bois with a lot of cream; while Bond shares the caviar starter before moving on to a very small tournedos, underdone, with sauce Béarnaise and a coeur d’artichaut, and then (surprisingly) half an avocado with a little vinaigrette for dessert. And champagne to drink, Bond thinks the Taittinger 45, though the sommelier tactfully suggests the Blanc de Blanc Brut 1943 might be more appropriate. Wow.

The pleasure of gambling

During the long scene in the casino where Bond battles Le Chiffre at baccarat, Fleming explains the rules with crystal clarity so that even a non-gambler like myself gets drawn into the exciting battle of wills. Not only explains what’s at stake, but conveys the enjoyment.

‘Bond had always been a gambler. He loved the dry riffle of the cards and the constant unemphatic drama of the quiet figures round the green tables. He liked the solid, studied comfort of card-rooms and casinos, the well-padded arms of chairs, the glass of champagne or whisky at the elbow, the quiet unhurried attention of good servants… He liked being an actor and spectator… Above all he liked it that everything was one’s own fault…’ (p.47)

The pleasure of BDSM sex

When, after pages of the very long card sequence, Le Chiffre finally loses all his money (and thus Bond’s mission is complete) Bond is fantastically relieved. He cashes his checks and tokens in at the casino bank, then takes Vesper for a drink. Coming down off the tense high of the card game, he imagines having sex with Vesper, but not vanilla sex; rather, sex which involves domination and pain, tears and ecstasy.

He wanted her cold and arrogant body. He wanted to see tears and desire in her remote blue eyes and to take the ropes of her black hair in his hands and bend her long body back under his. (p.98)

Later, he repeats the same feeling only that, knowing Vesper better, and having become more intrigued by her mystery, he imagines the sexual act with deeper intimacy and fervour.

And now he knew that she was profoundly, excitingly sensual, but that the conquest of her body, because of the central privacy in her, would each time have the sweet tang of rape. Loving her physically would each time be a thrilling voyage without the anticlimax of arrival. She would surrender herself avidly, he thought, and greedily enjoy all the intimacies of the bed without ever allowing herself to be possessed. (p.167)

Where it is clear from the context, and from the descriptions of his deepening feelings for Vesper, that the word ‘rape’ is used in a BDSM context, meaning the agreed, permissive role-playing of violent or aggressive domination and submission, which is designed to take its participants to higher levels of sensuality and intimacy. Emphatically not the backstreet violence, the ugly violation of actual real-world rape.

The pleasure of driving a fast car

But before this can happen, Vesper is kidnapped by Le Chiffre’s people and Bond gives chase in the Bentley. Although there are technical descriptions of rear-wheel drives and superchargers and so on, what comes over most in this car chase is the sheer physical pleasure Bond gets from driving a superbly engineered car with supreme skill.

The baddies wait on a bend and lace the road with anti-tyre nails so that Bond’s lovely Bentley crashes at high speed and he is pulled unconscious from the wreckage, taken in the baddy’s car, along with the unconscious Lynd, to an isolated farmhouse and there tortured by Le Chiffre.

The masochistic pleasure of being tortured

Le Chiffre cuts away the seat of a cane chair and has Bond tied naked to it so his genitals are hanging down through the gap. Le Chiffre then proceeds to beat his genitals very hard with a domestic carpet beater until Bond is unconscious with pain, covered in sweat and there is a pool of blood under the chair from his damaged body. As he does so Le Chiffre’s eyes look at him ‘almost caressingly’ (p.120).

Linked to the S&M vision of sex with Vesper, this gross torture scene is part of the sensual world the book inhabits, a world of physical pleasures and terrifying pains. The horrifying torture is the mirror image of the cold showers, the slick grooming, the smooth shaves and bow ties which precede the fine dinners.

It reaches its climax when Le Chiffre, mock sorrowful that Bond has not revealed to him the location of the vast cheque of his winnings (which, understandably, he wants to steal) takes out a big kitchen knife and advances on Bond to emasculate him. ‘Say goodbye to it, Mr Bond’ (p.127). Presumably this means Le Chiffre is going to cut Bond’s penis off.

Le Chiffre had earlier assured Bond that this wasn’t the kind of ‘romantic adventure’ in which the hero is rescued by magic. But in fact it is. As Le Chiffre advances to castrate Bond, a mysterious voice sounds in the gloom. It is an agent of SMERSH sent to terminate him for his stealing of the KGB funds, which Russia has, after all, found out about. ‘Phut’ goes the silenced gun and Le Chiffre falls dead. The SMERSH agent whispers in Bond’s ear that, unfortunately, he has no orders to kill him, but he sadistically carves the Russian character for the first letter of Spionam into his right hand, and departs.

The pleasure of recovery

Out of a fog of anaesthetics Bond surfaces in a French hospital. The police had found his crashed car, searched nearby houses and discovered him and Vesper with the corpses of Le Chiffre and his henchmen. The doctors are (of course) amazed at Bond’s superhuman powers of endurance and, indeed, recovery. Miraculously ‘he’ will heal – ie his testicles, which you would have thought would have been mashed to pieces, will be restored to good working order. What is more restful and pleasurable than lying in a hospital bed, doped with local anaesthetic, being ordered to do nothing, think of nothing, and just be fed and watered day and night?

Vesper’s betrayal

The novel could have ended about here, when Bond – surprisingly – tells Mathis, who’s come to visit him in hospital, that he’s going to quit his job. In a couple of pages of schoolboy philosophising he says it’s getting harder to tell the good guys from the bad guys. Well, don’t fight for principles, Mathis advises him; fight for the people you love and against the people who are threatening them.

Bond is worried he won’t have healed enough to have sex with Vesper (or anyone) so is sensitive when she visits him in hospital. After visiting every day for a week or more, Vesper says she has found a quiet little boarding house down the coast and arranged for them to stay. They drive there with basic belongings and begin what should be an idyllic beach holiday. There is a sensuous build-up with swimming from the deserted beach, an immaculate home-made French dinner, and then passionate love-making.

But almost immediately Vesper becomes tense and nervous. Next morning Bond catches her making a secretive phone call which she unconvincingly lies about. She is convinced someone has followed them to the boarding house, and when a passing commercial traveller stops for lunch, she tenses with fear.

They continue the days of sunbathing and nights of good food and sex, but Vesper cries half the time and is irreparably sad. There is one last night of love before which Vesper insists on getting tipsy, and then the terrified patron wakes Bond early the next morning (it is the respectable 1950s, so Bond and Vesper had been given separate rooms). Bond rushes down to the hall to find Vesper dead in her bed. She has committed suicide by taking an overdose of sleeping pills.

Next to her is a farewell letter in which she reveals that she has been a double agent working for the Soviet MWD for years, ever since the Russians got hold of her Polish RAF pilot lover. The Russians promised he’d be safe if she spied for them. And so for years – years, Bond reflects bitterly, during which he has been gallivanting round the globe like a schoolboy adventurer bumping off cartoon baddies – the real traitor has been working in the heart of his organisation, quietly copying top secret files and sending their contents to Moscow.

Vesper admits that she betrayed every detail of the Le Chiffre mission to Moscow, including Bond’s cover story and aims. In the casino, she distracted Leiter so Le Chiffre’s would-be assassin could come close to killing Bond, with a silenced concealed gun. And, she reveals, her kidnap by Le Chiffre was all a put-up job to entrap him. Vesper goes on to say she genuinely fell in love with Bond but didn’t want to betray her Polish lover, and when she stopped regularly messaging her Paris contact, she knew they would send someone for her, someone from SMERSH. Which explains her irrational fear of the ‘commercial traveller’. So now she’s taken the only way out of her hopelessly tangled, compromised plight.

The bitch is dead

Bond crumples up the letter, all sentiment for her evaporated. All this time she was a spy doing her country inestimable damage. And he had grand-standed in front of Mathis, saying he was ready to marry Vesper and quit the service because he was all confused about the morality of spying. Not any more.

The novel ends with his determination to combat the fear which stands behind all the Soviet agents, the whip hand of SMERSH itself, the instrument of terror which keeps the whole system running. He phones the hot line to London to leave a coded message, that agent 3030 was a double all along. Yes, he said ‘was’ – because ‘the bitch is dead now.’

If Bond was cold and heartless before, he is even colder and more heartless now. If he was wavering about his job and his role, this mission crystallises his determination to shake himself out of it, and take the fight to the enemy.

Thoughts

This is an excellent kick-start to what turned out to be a never-ending series of fictions about the cold, pitiless, sensual, cruel connoisseur spy. All the key ingredients are here, including the final determination to fight the foe forever. And Bond may indeed go on forever…

This first book establishes the narrative pattern: Bond combats one big, central baddy, preferably in exotic foreign locations, where he displays his connoisseur-like enjoyment of the finer things in life and survives numerous physical attacks, before the plot intensifies, he falls into the baddy’s clutches and endures sadistic levels of punishment, before just about defeating the enemy, all accompanied by a sexy lover whose initial coldness he triumphantly overcomes.


Credit

Casino Royale published in 1953 by Jonathan Cape. All quotes and references are to the 1978 Triad Grafton paperback.

Related links

Other thrillers of 1953

The Bond novels

1953 Casino Royale Bond takes on Russian spy Le Chiffre at baccarat then is gutted to find the beautiful assistant sent by London to help him and who he falls in love with – Vesper Lynd – is herself a Russian double agent.
1954 Live and Let Die Bond is dispatched to find and defeat Mr Big, legendary king of America’s black underworld, who uses Voodoo beliefs to terrify his subordinates, and who is smuggling 17th century pirate treasure from an island off Jamaica to Florida and then on to New York, in fact to finance Soviet spying, for Mr Big is a SMERSH agent. Along the way Bond meets, falls in love with, and saves, the beautiful clairvoyant, Solitaire.
1955 Moonraker An innocent invitation to join M at his club and see whether the famous Sir Hugo Drax really is cheating at cards leads Bond to discover that Drax is in fact a fanatical Nazi determined on taking revenge for the Fatherland by targeting an atom-bomb-tipped missile – the Moonraker – at London.
1956 Diamonds Are Forever Bond’s mission is to trace the route of a diamond smuggling ‘pipeline’, which starts in Africa, comes to London and then to follow it on to New York, and further to the mob-controlled gambling town of Las Vegas, where he wipes out the gang, all the while falling in love with the delectable Tiffany Case.
1957 From Russia, with Love Bond is lured to Istanbul by the promise of a beautiful Russian agent who says she’ll defect and bring along one of the Soviets’ precious Spektor coding machines, but only for Bond in person. The whole thing is an improbable trap concocted by head of SMERSH’S execution department, Rosa Klebb, to not only kill Bond but humiliate him and the Service in a sex-and-murder scandal.
1958 Dr. No Bond is dispatched to Jamaica (again) to investigate the mysterious disappearance of the station head, which leads him to meet up with the fisherman Quarrel (again), do a week’s rigorous training (again) and set off for a mysterious island (Crab Key this time) where he meets the ravishing Honeychile Rider and the villainous Chinaman, Dr No, who sends him through a gruelling tunnel of pain which Bond barely survives, before killing No and triumphantly rescuing the girl.
1959 Goldfinger M tasks Bond with finding out more about Auric Goldfinger, the richest man in England. Bond confirms the Goldfinger is smuggling large amounts of gold out of the UK in his vintage Rolls Royce, to his factory in Switzerland, but then stumbles on a much larger conspiracy to steal the gold from the US Reserve at Fort Knox. Which, of course, Bond foils.
1960 For Your Eyes Only (short stories) Four stories which started life as treatments for a projected US TV series of Bond adventures and so feature exotic settings (Paris, Vermont, the Seychelles, Venice), ogre-ish villains, shootouts and assassinations and scantily-clad women – but the standout story is Quantum of Solace, a conscious homage to the older storytelling style of Somerset Maugham, in which there are none of the above, and which shows what Fleming could do if he gave himself the chance.
1961 Thunderball Introducing Ernst Blofeld and his SPECTRE organisation who have dreamed up a scheme to hijack an RAF plane carrying two atomic bombs, scuttle it in the Caribbean, then blackmail Western governments into coughing up $100,000,000 or get blown up. The full force of every Western security service is thrown into the hunt, but M has a hunch the missing plane headed south towards the Bahamas, so it’s there that he sends his best man, Bond, to hook up with his old pal Felix Leiter, and they are soon on the trail of SPECTRE operative Emilio Largo and his beautiful mistress, Domino.
1962 The Spy Who Loved Me An extraordinary experiment: an account of a Bond adventure told from the point of view of the Bond girl in it, Vivienne ‘Viv’ Michel, which opens with a long sequence devoted entirely to her childhood in Canada and young womanhood in London, before armed hoodlums burst into the motel where she’s working on her own, and then she is rescued by her knight in shining armour, Mr B himself.
1963 On Her Majesty’s Secret Service Back to third-person narrative, and Bond poses as a heraldry expert to penetrate Blofeld’s headquarters on a remote Alpine mountain top, where the swine is carrying out a fiendish plan to use germ warfare to decimate Britain’s agriculture sector. Bond smashes Blofeld’s set-up with the help of the head of the Corsican mafia, Marc-Ange Draco, whose wayward daughter, Tracy, he has fallen in love with, and in fact goes on to marry – making her the one great love of his life – before she is cruelly shot dead by Blofeld, who along with the vile Irma Bunt had managed to escape the destruction of his base.
1964 You Only Live Twice Shattered by the murder of his one-day wife, Bond goes to pieces with heavy drinking and erratic behaviour. After 8 months or so M sends him on a diplomatic mission to persuade the head of the Japanese Secret Service, ‘Tiger’ Tanaka to share top Jap secret info with us Brits. Tiger agrees on condition that Bond undertakes a freelance job for him, and eliminates a troublesome ‘Dr Shatterhand’ who has created a gruesome ‘Garden of Death’ at a remote spot on the Japanese coast. When Bond realises that ‘Shatterhand’ is none other than Blofeld, murderer of his wife, he accepts the mission with gusto.
1965 The Man With The Golden Gun Brainwashed by the KGB, Bond returns from Japan to make an attempt on M’s life. When it fails he is subjected to intense shock therapy at ‘The Park’ before returning fit for duty and being dispatched to the Caribbean to ‘eliminate’ a professional assassin, Scaramanga, who has killed half a dozen of our agents as well as being at the centre of a network of criminal and political subversion. The novel is set in Bond and Fleming’s old stomping ground, Jamaica, where he is helped by his old buddy, Felix Leiter, and his old secretary, Mary Goodnight, and the story hurtles to the old conclusion – Bond is bettered and bruised within inches of his life – but defeats the baddie and ends the book with a merry quip on his lips.
1966 Octopussy Three short stories in which Bond uses the auction of a valuable Fabergé egg to reveal the identity of the Russians’ spy master in London; shoots a Russian sniper before she can kill one of our agents escaping from East Berlin; and confronts a former Security Service officer who has been eaten up with guilt for a wartime murder of what turns out to be Bond’s pre-war ski instructor. This last short story, Octopussy, may be his best.

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