John Allen Paulos, INNUMERACY: Mathematical Illiteracy and Its Consequences (1988)

This slender book was first published in 1988 (I have the 1990 paperback edition) and became a bestseller. It was one of the earliest books on the very general theme of how many people don’t understand the world around them, or believe things about it that aren’t so. (This is what I mean, partly, when I refer to people who “don’t know how the world works,” in the sense they have no idea of proportion and size and thus no clue about what things are plausible or not – and thus are prey to so many conspiracy theories.)

This theme has been greatly extended in the past three decades, more generally about the many psychological biases people are prone to, as well as about the specific crazy things they believe. Gilovich’s HOW WE KNOW WHAT ISN’T SO came along in 1991; the first Shermer came along in 1997; and then a whole bunch of books beginning in the 2000s.

(Anticipating them was the textbook I’ve written about, LOGIC AND CONTEMPORARY RHETORIC, first published in 1971, but that wasn’t quite the same theme. Similarly books like Martin Gardner’s 1957 FADS & FALLACIES IN THE NAME OF SCIENCE and other books about pseudoscience and flimflam, some of which go back decades about phenomena that have been around for centuries; it too is an adjacent theme.)

I first read INNUMERACY in 1992, and revisited it recently to capture these notes. More than I’d remembered, the themes here aren’t strictly about math knowledge as they are techniques of thinking, or reasoning, in much the way so many of those later books have been about.

As with many of Paulos’ books, this one is easy to summarize at a chapter level, but even one level below that, it consists of lots and lots of specific examples. So I’ll leave my raw notes, with all those examples, fairly intact, and extract the big themes as key points.

Key Points

  • There are examples of innumeracy in daily life, from not understanding the difference between million and billion, to the inability to estimate really big things. Basic principles people should understand include those of combinatorics, probabilities, and percentages.
  • Many counter-intuitive events (“coincidences”) can be understood through calculation of probabilities. It would be unlikely for the occasional unlikely event *not* to occur. Applications to birthdays, chance encounters, the stock market, choosing a spouse, tossing coins, the “hot hand.”
  • Math can be misused to promote pseudoscience: astrology, I Ching; biorhythms. Freud, Marxism, UFOs–claims that are largely unfalsifiable. Predictive dreams. Medical scams: psychic surgeons, TV evangelists. Numerology.
  • What can be done about innumeracy? Grade school teachers could be better taught. More mathematicians encouraged to write for general audiences. Understand the bias in news coverage toward coincidences. Depersonalize math (mathematicians, he notes, have quirky senses of humor) and the link between math and puzzles could be highlighted. And use math more intelligently, e.g. by applying the unit pricing idea to risk assessment, as in the probability of dying by a terrorist attack compared to that of slipping in your bathtub.
  • Understand societal impacts of using math, in areas of voting, type I and type II errors, polling, doing randomly selected sampling, avoiding alarming prognoses; beware averages in favor of ranges; understand that much of life involves trade-offs, e.g. of quality v price.

One of the author’s recurring themes is how humans perceive meaning in things that are actually random, even if they are subject to probabalistic calculation. Quote, p178.4:

…giving due weight to the fortuitous nature of the world is, I think, a mark of maturity and balance. Zealots, true believers, fanatics, and fundamentalists of all types seldom hold any truck with anything as wishy-washy as probability.

At the end the author summarizes his reasons for writing the book – how a society so dependent on science on math is so indifferent to literacy of them; how we spend so much on a military with so poorly educated soldiers; how the media favors the boy-down-the-well story over bigger problems of urban crime, poverty, and the environment, that take mathematical analysis to appreciate. And the “rampant silliness” of astrology and other pseudoscience…

Still, irritation with these matters was only part of my incentive. The discrepancies between our pretensions and reality are usually quite extensive, and since number and chance are among our ultimate reality principles, those who possess a keen grasp of these notions may see these discrepancies and incongruities with greater clarity and thus more easily become subject to feelings of absurdity. I think there’s something of the divine in these feelings of our absurdity, and they should be cherished, not avoided. They provide perspective on our puny yet exalted position in the world, and are what distinguish us from rats. Anything which permanently dulls of to them is to be opposed, innumeracy included. The desire to arouse a sense of numerical proportion and an appreciation for the irreducibly probabilistic nature of life—this, rather than anger, was the primary motivation for the book.



  • Author cites examples of innumeracy in casual conversation, and how pundits at grammar don’t care about math. This book will focus on many examples from everyday life. Innumeracy tends to link with belief in pseudoscience. People tend to be misled by their own experiences.

Ch1, Examples and Principles

  • Author is amazed by various examples of innumeracy. Examples: How many people haven’t the vaguest idea about the population of US (other examples p8); Have an ability to estimate; Understand the relative dangers of, say, auto accidents vs terrorist attacks; Know the difference between million, billion, trillion. (With an aside about scientific notation.)
  • Examples of big numbers used and abused: The number of ways to solve Rubik’s cube; How long a trillion seconds is, compared to numbers in federal budget; Try estimating really big things, 14t. Examples: the volume of all human blood in the world. Snails vs Concordes. How long for dump trucks to move Mount Fuji. (Answer: five to ten thousand years).
  • Examples of how authors across history have been inconsistent in their use of large numbers. Gargantua; the Genesis flood. A bit of numeracy can allow dismissal of some claims on the basis of a few raw numbers. You can’t scale up a man, as with Gargantua.
  • Principles, beginning with Archimedes: any big number can be exceeded by adding together sufficiently many smaller numbers. Thus his fulcrum. He calculated the number of grains of sand it would take to fill the earth and the heaves, p20. Similar calculations now.
  • Next, some basic multiplication principles. Combinatorics: the number of ways to roll a pair of dice, etc. Mozart wrote a waltz allowing for different possible arrangements of notes. Most people don’t realize how great such numbers are. Baskin Robbins 31 flavors; how many combinations of 3? Lotteries. Poker hands.
  • Probabilities, e.g. of successive independent events, like tossing coins. Examples of things occurring and not occurring (1 – occurring). A trick to use a biased coin to get fair results, p29-30. Probability distributions, p30.
  • We can use percentage chances of *not* catching this or that disease, but the chance of catching any one of them might still be rather high. How likely a deep breath contains atoms breathed by Caesar? About 99%. Author describes the calculation.

Ch2, Probability and Coincidence

  • Freud thought nothing was coincidence; Jung spoke of synchronicity. Everyone speaks of ironies. But they are all much more common than people realize. Thus innumerates accord them great significance. The example of how many people needed for two of them to have the same birthday; author derives the answer, for 50%, which is 23. *Some* unlikely events occur frequently; any particular event, rarely. It would be unlikely for unlikely events not to occur. This is how psychics, whose predictions are sufficiently vague, work.
  • Similarly for chance encounters. How short the chains are between any two people. How author once wrote to Bertrand Russell. Examples of hats, letters, shuffled cards. The Dirichlet drawer principles.
  • Example of the stock market con man who sends follow-ups only to those whose predictions he sent them came true. People remember the successes but not the losses; thus casinos. [[ Note author calls this a “filtering phenomenon” but doesn’t use the modern terminology … ]] Similarly how ordinary and local events are never as extraordinary as all-time or international events. Thus local entertainers never satisfy; everyone has been spoiled by the world-class.
  • Expected values are more informative than extreme values; how insurance works. How medical clinic tests can be done. The game of chuck-a-luck, 48-49.
  • Choosing a spouse: one could calculate the chances. The policy works out to be: estimate the number you might ever meet, then pass up the first 37% and choose the next heartthrob that turns up.
  • Coincidence and the law: example of thief and driver descriptions; how many potential matches? What is probability all descriptors would match?
  • Any particular combination of number (e.g. on a lottery ticket) is improbable; so why do people like certain ones and not others? 54m. Baseball players. A man with two girlfriends.
  • Counter-intuitive results, e.g. tossing coins, p57. Each toss is independent; a streak won’t cause a ‘return’ to average. Thus people can be dubbed winner or loser at random. Random events can seem quite ordered, without ‘explanation’. Thus stock market analysts always provide ‘answers’ to trends when none necessarily exist.
  • Hot hands: ref Tversky and Kahneman. P61. They discredited the idea of a ‘hot hand’. Other events, like numbers of children in a family, can be similarly anticipated. Even quite rare events are thus predictable.

Ch3, Pseudoscience

  • Math deals with certainties, but it can be misused when assumptions are misapplied, 67. Substitutions. Old nonsense can be computerized but that doesn’t make it any less nonsense. Astrology, I Ching; biorhythms. Even Freud believed this stuff. Freud’s claims were largely unfalsifiable. As Marxism. Similar sloppy thinking about UFOs. Some statements are just vacuous, p70.
  • Parapsychology. No repeatable studies have verified its existence. General observations: the burden of proof is on those who claim it. An incident of ESP looks just like a chance guess. There is tremendous will to believe. People remember a few hits and forget the many misses; supermarket tabloids. For a while phrenology was the thing. Fire walking. For many others: see The Skeptical Inquirer.
  • Predictive dreams. They are accounted for by coincidence. Do the odds. Even with low probabilities, there are millions of such dreams every year, across the population. Similarly other kinds of incredible coincidences.
  • Astrology. A dismaying number of people believe in it; some 52% in a 1968 poll. But there is no plausible mechanism for it work. (This doesn’t concern the innumerate.) Also, there’s no empirical evidence that it works. Why then? The predictions are so vague. The flattering implication that one matters in the vast scheme of things. And individual astrologers are skilled at reading their clients.
  • ET life, UFO visitors; yes and no. Again numeracy might help. Numbers of stars, of planets that might support life; and the size of the galaxy. But how civilizations are likely scattered in time, and why would they be interested in us?
  • Medical treatments. Fraudulent claims. Again, claim the hits and explain away the misses. Many areas: psychic surgeons, TV evangelists, etc. (Not to advocate … “some kind of simpleminded atheism” author stresses, 83.8.) Quine argues that no experience ever forces one to reject any particular belief, 84.6.
  • Conditional probability, etc. Mistakes in reasoning, probability of A given B. Examples. Con game with red and black cards. Only in blackjack does it make sense to keep track of the past. Bayes’ theorem. Drug and AIDS testing; cancer tests.
  • Numerology. Ancient practices. Numerical values for each letter. Long history of examples. There are always ways to come up with whatever conclusion you want.
  • Logic and pseudoscience. Common are confusing conditional statements. Examples of silly conclusions. Lack of evidence doesn’t mean something is true.

Ch4, Whence Innumeracy?

  • Why so widespread among otherwise educated people? Poor education, psychological blocks, and romantic misconceptions about the nature of mathematics.
  • Example from author’s childhood; the teacher was wrong. In school, even simple applications are rarely taught. Estimation is rarely taught. Nor inductive reasoning. Nor the connection to puzzles. Teachers are not well trained. Mathematicians themselves often have weird senses of humor.
  • High school and beyond. Many students still have troubles with algebraic variables, with Cartesian plots. High school students should be exposed to the most important ideas in finite math. Grad study is best. Few mathematicians write for general audiences.
  • Psychological factors: the tendency to personalize. While math seems impersonal.
  • Filtering and coincidence. Psychology, broadly speaking—how we filter the world. The bias in news coverage. And so many coincidences, it seems. We have innate desires for meaning and pattern. The regression to the mean. It just happens; there is no law that it must happen. Many examples.
  • Decisions and framing questions. They bank teller lady question, p115. Here Judy. Or the same question framed two different ways.
  • Math anxiety. Techniques to counter this include explaining the problem to someone else; using more relatable numbers. And there’s an issue of student lethargy.
  • Romantic misconceptions. Yes, math is cold in some ways; but it relates to the real world. Mathematicians are as subject to human motivations as anyone else. In some ways numbers can depersonalize. Or can seem limiting. (as can reality.) or can seem deterministic—until Godel.
  • Digression about supermarket unit pricing. Consider similar safety, or danger, indexes to allow comparison of different activities. Logarithmic. Risks, diseases, accidents. A statistical ombudsman for the media would be a good idea.

Ch5, Statistics, Trade-Offs, and Society

  • Examples of bone-headed statistics.
  • This is about the harmful effects of innumeracy, especially individual vs society.
  • Consider example of a game of four dice, that’s analogous to the paradoxes of most voting systems. Individual rationality can lead to social irrationality. Preferential voting. Four conditions for voting, 136.8.
  • Laissez-faire. A philanthropist dilemma with Russian roulette; examples of exchanges of any sort (whether or not to dupe the other guy); the prisoner’s dilemma. In these cases cooperation is the best policy.
  • Birthdays…  probability and statistics, both from the 17th century. Interesting case of whether death-days correlate with birthdays; perhaps they do. Another example concerning ESP.
  • Type I and Type II errors. Liberals and conservatives tend in different ways. [[ significant point? It’s about comfort for risk and uncertainty. ]] trade-offs in manufacturing.
  • Polling. The size of the sample means a lot. Examples. Margins of error are important. And it can be difficult to select ‘random samples’ e.g. problems with telephone polls. Need to avoid biased results. Be careful of self-selected samples.
  • Personal Information. How to obtain small randomly selected samples? Examples: flip a coin to mean truth or lie; the capture-recapture method.
  • Two theoretical results. The law of large numbers: in the long run, the probability of an event and its frequency converge. The central limit theorem: a large bunch of measurements follows a bell curve.
  • Correlation and causation. Examples. Confounding factors. Accidental correlations. Regression analysis.
  • Simple mistakes. Likelihood of breast cancer in women. Percentages going up and down; always ask, percentage of what? And fractions. Eyewitness evidence. Women’s salaries vs men’s; the figure cited lacks all context. Similarly, the percentage of the population that owns the percentage of the wealth. Even adding can be mistaken. How children have no time for school. Or quoting an absolute number rather than a probability; probability puts things in perspective.
  • Odds and Addenda. Beware the urge to average. They don’t tell you about ranges. Various games. Etc etc. “Statistical significance” often doesn’t mean much. We often deny necessary trade-offs. E.g. quality and price, etc. These trade-offs imply a finite economic value on a human life.

Close, p177.

  • An appreciation for probability takes a long time to develop. “In fact, giving due weight to the fortuitous nature of the world is, I think, a mark of maturity and balance. Zealots, true believers, fanatics, and fundamentalists of all types seldom hold any truck with anything as wishy-washy as probability.” P178.4.
  • Part of author’s motivation for this book is anger. With the mismatch in spending, with what the media focus on, with the rampant silliness of astrology et al.
  • And finally: appreciating the discrepancy between our pretensions and reality. Last para (see above).
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