Nature’s Numbers by Ian Stewart (1995)

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

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

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

1. The Natural Order

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

2. What Mathematics is For

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

Stewart trots through the history of major mathematical discoveries: Kepler discovers that the planets move not in circles but in ellipses. That the nature of acceleration is ‘not a fundamental quality, but a rate of change’. Then Newton and Leibniz invent calculus to help us work out rates of change.

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

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

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

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

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

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

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

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

3. What Mathematics is About

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

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

The invention of numbers. Fractions. Some time in the Dark Ages the invention of 0. The invention of negative numbers, then of square roots. Irrational numbers. ‘Real’ numbers.

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

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

Maths is like a landscape with similar proofs and theories clustered together to create peaks and troughs.

4. The Constants of Change

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

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

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

5. From Violins to Videos

A fascinating historical recap of how initial investigations into the way a violin string vibrates gave rise to formulae and equations which turned out to be useful in mapping electricity and magnetism, which turned out to be aspects of the same fundamental force, the understanding of which underpinned the invention of radio, radar, TV etc – taking in descriptions of the contributions from Michael Faraday, James Clerk Maxwell, Heinrich Hertz and Guglielmo Marconi.

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

6. Broken Symmetry

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

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

7. The Rhythm of Life

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

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

8. Do Dice Play God?

Stewart’s take on chaos theory.

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

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

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

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

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

9. Drops, Dynamics and Daisies

The book ends by drawing a kind of philosophical conclusion.

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

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

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

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

In all three cases the underlying principles seem to be resolvable into easily stated laws and functions – and we see water dropping off taps or flowerheads all the time – and yet the intermediate steps between principle and real world embodiment, are mind-bogglingly complex.

Coda: Morphomatics

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


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Maths ideas from John Allen Paulos

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Prisoner’s dilemma graphic. Source: Wikipedia

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

and defined by the formula:

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

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

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

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

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

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


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

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

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

John Allen Paulos

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

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

Structure

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

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

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

Response

The book is disappointing in all kinds of ways.

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

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

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

Nope. Not at all.

Lani ‘Quota Queen’ Guinier

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

Why not?

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

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

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

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

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

So:

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

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

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

This scattergun approach characterises the whole book.

Psychological availability and anchoring effects

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

General points

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

  • almost all ‘news’ reinforces conventional wisdom
  • because they’re so brief, almost all headlines must rely on readers’ existing assumptions and prejudices
  • almost all news stories relate something new back to similar examples from the past, even when the comparison is inappropriate, again reinforcing conventional wisdom and failing to recognise the genuinely new
  • all economic forecasts are rubbish: this is because economics (like the weather and many other aspects of everyday life) is a non-linear system. Chaos theory shows that non-linear systems are highly sensitive to even minuscule differences in starting conditions, which has been translated into pop culture as the Butterfly Effect
  • and also with ‘futurologists’: the further ahead they look, the less reliable their predictions
  • the news is deeply biased by always assuming human agency is at work in any outcome: if any disaster happens anywhere the newspapers always go searching for a culprit; in the present Brexit crisis lots of news outlets are agreeing to blame Theresa May. But often things happen at random or as an accumulation of unpredictable factors. Humans are not good at acknowledging the role of chance and randomness.

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

  • Hence so many newspapers endlessly playing the blame game. The Grenfell Tower disaster was, first and foremost, an accident in the literal sense of ‘an unfortunate incident that happens unexpectedly and unintentionally, typically resulting in damage or injury’ – but you won’t find anybody who doesn’t fall in with the prevailing view that someone must be to blame. There is always someone to blame. We live in a Blame Society.
  • personalising beats stats, data or probability: nothing beats ‘the power of dramatic anecdote’ among the innumerate: ‘we all tend to be unduly swayed by the dramatic, the graphic, the visceral’ (p.82)
  • if you combine human beings’ tendency to personalise everything, and to look for someone to blame, you come up with Donald Trump, who dominates every day’s news
  • so much is happening all the time, in a world with more people and incidents than ever before, in which we are bombarded with more information via more media than ever before – that it would be extraordinary if all manner or extraordinary coincidences, correspondences and correlations didn’t happen all the time
  • random events can sometimes present a surprisingly ordered appearance
  • because people imbue meaning into absolutely everything, then the huge number of coincidences and correlations are wrongfully interpreted as meaningful

Tips and advice

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

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

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

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

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

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

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

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

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

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

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

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

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

But just occasionally he does say something unexpected:

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

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

Complexity horizon

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

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

Social history

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

Thus Paulos has essays on:

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

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

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

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

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

Something bad is coming

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

‘Imagine…’

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

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

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

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

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

‘Assume that…’

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

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

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

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

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

Jokes and Yanks

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

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

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

The moronic inferno and the liberal fallacy

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

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

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

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

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

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

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

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

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

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

‘An informed and effective citizenry’?


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