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Innumeracy : Mathematical Illiteracy and Its Consequences ... John Allen Paulos

Innumeracy : Mathematical Illiteracy and Its Consequences ... John Allen Paulos

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    Innumeracy Innumeracy Document Transcript

    • Innumeracy Index “broad base” fallacy, 125. See also statistics, typical A misinterpretations of absolute numbers, 44 burden of proof, 52 accidental death, 121–122 Burns, Marilyn (I Hate Mathematics), 75 Adam Smith’s “invisible hand,” 104 additivity, concept of (Archimedes), 14, 15 C AIDS, 21–22 Cabala, 90 algebra, 77–78 cabalism. See numerology anecdotes, 114 calculation, 9–13, 15–23 anxieties (regarding numbers), 4–5 of probabilities, 7–20 applied science, 105 calculus, 38, 78 Archimedean property of numbers, 14–15 capture-recapture method, 115–116 Archimedean question, 16 Carlson, Shawn, 57 Archimedes, 14–15 casinos, 33, 36. See also gambler’s fallacy; lotteries; fulcrum claim of, 15 slot machines Arrow, Kenneth, 100–101 celebrities, 30 Asimov, Isaac, 49 Cent mill milliards de poems, 18 astrology, 26, 28, 49, 50, 53, 56–59 central limit theorem, 117–118 experiments with, 57 chance persistence of belief in, 58 and AIDS, 21–22 atheism, 62 in bridge, 40 attitude, 4, 5 and correlation, 120–121 averages, 33–34, 41, 117–118. See also bell-shaped and death, 23 curve; mean and disease, 23–24, 121–122 batting, 41, 47 encounters, 29 and blood tests, 35–36 and Julius Caesar, 24 and coin flips, 43–46 and law, 39–41 and discrimination, 41–42 and love, 37–39 and love, 42 and “meaning,” 83 perceptions of, 34, 126–127 and the stock market, 85, 128 Axelrod, Robert, 105 and success, 33 axioms, 78 chuck-a-luck, 36–37. See also binomial probability distribution B Church, Alonzo, 93 Bacon, Francis, 114 Clever Hans (horse), 58 baseball. See batting averages “clutch hitter.” See batting averages basketball. See consecutive baskets coefficients, combinational, 20 batting averages, 41, 47 coin flips, 43–46, 114–115, 128 Bayes theorem, 66 coincidence, 5, 25–31, 34, 37, 39–40, 83. See also bell-shaped curve, 117–118, 127. See also averages; rarity mean and acquaintance, 29–30 Bernoulli, Jean, 116 frequency of, 6, 26, 40 betting, 64–65, 128 and innumeracy, 26 bias, 21, 81–83 and law, 39–41 billions, 8, 10 and lotteries, 41 binomial probability distribution, 22, 36–37, 40, 48, combinational coefficients, 20 106–109. See also chuck-a-luck combinatorics. See mathematics, finite biorhythms, 50–51 computer, universe compared to, 16 birthdays, 106 conditional probabilities, 38–39, 85–87, 100–101 blackjack, 65–66 conditional probability, 63–66. See also probabilities blood tests, 35–36 Condorcet, Marquis de? 100–101 Bloom, Allan, 4 confidence intervals, 111-116, 121-126 bridge (game), 40 consecutive baskets, 46 Created by K. Harley - 1
    • Index cooperation, 103–105 correlation versus causation, 118–121 F correspondences. See coincidence fads counting cards, 65–66 birth of, 33 Cowper, William, 49 as “evidence,” 53 “creation science,” 56 faith healing. See television evangelism CSI (Committee for Skeptical Inquiry), 54 fallacies, 70–71 CSICOP (Committee for the Scientific Investigation of falsification principle, 51 Claims of the Paranormal). See CSI (Committee for FDA (Federal Drug Administration) approval, 108– Skeptical Inquiry) 109 filtering phenomenon, 33–34, 53, 58, 61–62, 81–83, D 114 danger, perceptions of, 94–98 firewalking, 54 De Mère, Chevalier, 21 Fisher, R.A., 114 death, 23 Fliess, Wilhelm, 50 depression, 82 Flood, the, 13 Descartes, 77 flying saucers, versus UFOs, 51 Dirichlet drawer principle, 31. See also pigeonhole fractions, 122–123 principle framing, 86–87 discrimination, 41–42 fraud, 53, 64–65. See also pseudoscience; stock disease. See chance and disease market scams disputed authorship, 116 medical treatments, 61–62 Dixon, Jeane, 52–53 free lunch, 110, 111 dreams (predictive). See precognition Freud, Sigmund, 25, 49–51 Freudianism, 51 E fulcrum, 15 economics, 131–132 education, 73–80 G Educational Testing Service, 78 galaxy (Milky Way), size of, 59–60 Efron, Bradley, 100 gambler’s fallacy, 43–44, 84–85, 117. See also “error curve.” See bell-shaped curve lotteries; slot machines; casinos errors game theory. See mathematics, finite sampling, 111 Gardner, Martin, 28, 74, 80 Types I and II, 108–109 Gauss, 79 ESP, 52–53, 71, 107, 113. See also precognition Gauss, Carl Friedrich, 117 estimation, 7, 11, 13, 74, 110–113, 114–116, 121 Geller, Uri, 52 of commerce, 26 Genesis, 13 Euler, 79 geometry, 77–78 events Gödel, Kurt, 90, 93 general vs. specific, 27–28 Gombaud, Antoine. See De Mère, Chevalier unlikely, occurrence of, 28, 34 graph theory. See mathematics, finite events, dependence of, 85–87 Greeley, Horace, 53 events, independence of, 20–21, 47–48, 63, 65, 85. guesses, 52 See also independence of events principle defined, 20 H events, rarity of, 82 Hardy, G. H., 6 evidence, accumulation of, 83, 114 heredity, 83–84 evolution, 56 Hobbes, Thomas, 105 expected values, 33, 34–36, 55–56. See also value Hofstadter, Douglas, 9, 80 experience, 62 Homo sapiens, 10 exponential notation. See scientific notation Hopkins, Budd (Intruders, The), 59 extrasensory perception. See ESP “hot hand.” See consecutive baskets extraterrestrial beings, 51, 59–61 Huff, Darrell, 124 2
    • Innumeracy Index humor (of mathematicians), 76–77 M hypotheses testing, 105–106, 121 MacLaine, Shirley, 59 magical thinking, 131 I Mann, Horace, 53 I Ching, the, 50 Marquis de Condorcet, 100–101 improbabilities, 40–41 Marxism, 51 independence of events principle, 20–21. See also mass media, 97–98, 113 events, independence of math anxiety, 87–89 defined, 20 mathematical oddities, 25–42 individual versus society, 100–105 mathematicians, 76–77, 90 inductive reasoning, 74, 114–116 mathematics innumeracy, 114 and career options, 79 and anxiety, 87–89 and communication, 88–89 and comparison, 85–87 and constraints, 91 causes of, 4, 72–73, 83, 87–88 and context, 90 consequences of, 4, 7–8, 15, 50, 81–82, 99–132, deceptive use of, 49–52 118–132, 134–135 education, 73–80 defined, 3 finite, 78 examples of, 6–9, 12–14, 19 and graduate students, 79–80 in the numerate, 86 impersonal nature of, 80–82 and pseudoscience, 4–5, 49–50 and individuality, 90–92 psychological factors of, 80 and mental discipline, 89 and women, 88 and modeling, 129–130 innumeracy-of-the-year award, 120 popular perception of, 73, 80, 89–94, 122, 134 insurance, 28, 34–35 pure and esoteric, 50 intuitive appeal, 116 simplifying, 88 software, 75 J teacher competence in, 75–77 Judgment under Uncertainty, 86 and values, 93, 134–135 Jung, Carl, 25 and women, 78–79, 88, 123–124, 131 mathematics writers, 79, 80 K McGervey, John, 57 Kahneman, Daniel, 46, 84, 86–87 mean. See also averages; bell-shaped curve Kruth, Donald, 16 deviations from, 43 Kurtz, Paul, 54 regression to, 44, 83–84 meaning, 83 L medical tests, 35–36, 66 Laplace, 79 medical treatments, 70 law, 39–41 fraudulent, 61–62 law of large numbers, the, 116–117 Milgrim, Stanley, 29 lie-detection tests, 67 millions, 8, 10 light, speed of, 16 “mind over matter,” 54 literalism, 76 Molière, 51 logarithms, 94–95 monkey at typewriter. See “creation science” logic, 69, 74, 104 Mozart, Wolfgang Amadeus, 18 loopholes, 62 multiplication principle, 17, 21–24, 91 lotteries, 41. See also gambler’s fallacy; slot machines; casinos N love, 37–39, 42 Neanderthal, 10 heart versus head, 37 Nemerov, Howard, 62 ranking possible spouses, 37–39 New Age beliefs, 81 luck, 44 Newsweek, 120 normal distribution, 117–118 Created by K. Harley - 3
    • Index nuclear weapons, 10 of girls versus boys, 48 null hypothesis, 105–106 and law, 39–41 number anxiety, 4–5, 90–93 and lotteries, 41 numbers, Archimedean property of. see Archimedean and love, 37–39 property of numbers personalizing of, 5, 8, 44–45 numeracy, 9–12 and precognition, 55 defined, 13, 15 problems of, 21, 29–31, 35–36 numerology, 67, 67–69, 90 and randomness, 30–31, 35, 43–45 relation to statistics, 105–108 O probability. See mathematics, finite obscurantism, 94 probability theorems, 116–118 Overcoming Math Anxiety, 87–88 probability theory, 111, 116 pseudorandom numbers, 130 P pseudoscience, 98, 134. See also fraud; stock market paradoxes scams prisoner’s dilemma, 102, 103–105 and astrology, 56–59 Wolf’s dilemma, 102–103 defined, 58–59 St. Petersburg paradox, 128 and Freud, 50–51 parapsychology, 52–53 and innumeracy, 4–5, 49–52 Pascal, Blaise, 21, 93 mathematics in, 49–52 Pascal’s Wager, 107, 109–110 medical, 28, 33 pattern-finding tendencies, 83 and numerology, 67–69 Pearson, Karl, 114 persistence of belief in, 62 percentages, 3, 122–123 predictions of, 28 personal information, 114–116 versus science, 63 personality tests, 57 Pythagoras, 90 personalizing, 80–82 phrenology, 53 Q pigeonhole principle, 31. See also Dirichlet drawer quantum mechanics, 133 principle Queneau, Raymond, 18 Platonists, 90 Quine, Williard Van Orman, 62–63 Poisson probability distribution, 48 polls, 110, 128-129 R Dewey versus Truman, 113 Rabelais (Gargantua), 12–13 Landon versus Roosevelt, 113 Ramanujan, 6 Polya, George (How to Solve It and Mathematics and random sampling, 110–113 Plausible Reading), 74 random success. See stock market scams Popper, Karl, 51 randomness, 35, 43–45, 106–107, 115–116, 128–130, possible outcomes. See multiplication principle 133. See also probabilities “practical infinity,” 15 patterns within, 45–48 precognition, 54–56. See also ESP and probabilities, 30–31 prisoner’s dilemma, 102, 103–105 rarity, 40. See also coincidence privacy, 114 ratios, 44 probabilities, 20–24, 40, 46–47, 85–87, 102, 105, 106– regression analysis, 120–121 107, 111–113, 133. See also randomness revisionism, 51 appreciation of, 133–134 Rhine, J. B., 53 in basketball, 46 Richter scale, 94 and batting averages, 41 risks, 8, 87 binomial, 22 doctors’ assessment of, 8 calculation of, 7–20 scientists’ assessment of versus public’s, 4–5 conditional, 38–39, 85–87 Robertson, Pat, 70 and ESP, 52 Rousseau, Jean, 89 and evolution, 56 Rubik’s cube, 9 4
    • Innumeracy Index Russell, Bertrand, 30, 90 synchronicity, 25 Russian Roulette, 65, 102 T S “target individual” study, 29–30 safety index, 94–98 and celebrities, 30 Safire, William, 97 television evangelism, 28, 33, 62. Sagan, Carl, 58 textbooks, 75 Sahl, Mort, 101 theorems, 78 sampling, 128–129 thousands, 10–11 scale, 16 time units, 16 examples of, 9–14, 16 Tobias, Sheila, 87–88 inconsistency in, 12–14 trade-offs, 108–110 science, 58, 62, 83, 94 train schedules, 42 applied, 105 transitivity, 101 science, as defined by the innumerate, 81 trillions, 8, 10 Scientific American, 9 true value, 118 scientific notation Turing, Alan, 93 defined, 8–9 Tversky, Amos, 46, 84, 86–87 examples of, c, 15, 16 self-selection, 113 U separation, degrees of. See coincidence UFOs, 51, 61 Silverman, Bernard, 57 unidentified flying objects. See UFOs Skeptical Inquirer, The, 54 unit pricing, 94 skepticism, 7 universe slot machines, 33, 35. See also casinos; gambler’s age of, 16 fallacy; lotteries computer compared to, 16 Smith, Adam, 104 size of, 15–16 Smullyan, Raymond, 80 unknowns, 77 Soal, S.G., 53 social irrationality, 100–101 V social sciences, 120 values. See also expected values, average versus software, 75 extreme, 33–34 speed ratios, 12 Van Dumholtz, 70 St. Petersburg paradox, 128 Vieta, Franciscus, 77 statistical significance, 130–131 von Neumann, John, 22 statistics, 50, 52–53, 114–116 voting paradoxes, 100 and accounting, 124–125 and context, 124–126 W relation to probabilities, 105–108 Wall Street Journal, 129 types of, 105 Whitman, Walt, 72, 94 typical misinterpretations of, 121–126. See also Wolf, Robert, 102 “broad base” fallacy Wolf’s dilemma, 102–103 stock market, 85 advisors, 32 forecasts, 32–33 post hoc analysis of, 45 scams, 32–33 Strategic Defense Initiative (SDI), 108–109 Strieber, Whitley (Communion), 59 studies, faulty, 120–121 success, chances of, 33 sum, 117, 118, 125 symmetry, 48 Created by K. Harley - 5