How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843)

by Ellenberg, Jordan

  Only 47% of Americans: These numbers are from the January 2011 CBS poll cited above.

  In an October 2010 poll of likely voters: “The AP-GfK Poll, November 2010,” questions HC1 and HC14a, http://surveys.ap.org/data/GfK/AP-GfK%20Poll%20November%20Topline-nonCC.pdf.

  “The clause seems to express a great deal of humanity”: Annals of the Congress of the United States, Aug. 17, 1789. (Washington, DC: Gales and Seaton, 1834), 782.

  “The Court pays lip service”: Atkins v. Virginia, 536 US 304 (2002).

  Akhil and Vikram Amar: “Akhil Reed Amar and Vikram David Amar, Eighth Amendment Mathematics (Part One): How the Atkins Justices Divided When Summing,” Writ, June 28, 2002, writ.news.findlaw.com/amar/20020628.html (accessed Jan. 14, 2014).

  Just over six hundred people in all: Numbers of executions taken from the Death Penalty Information Center, www.deathpenaltyinfo.org/executions-year (accessed Jan. 14, 2014).

  you can train a slime mold to run through a maze: See, e.g., Atsushi Tero, Ryo Kobayashi, and Toshiyuki Nakagaki, “A Mathematical Model for Adaptive Transport Network in Path Finding by True Slime Mold,”Journal of Theoretical Biology 244, no. 4 (2007): 553–64.

  Tanya Latty and Madeleine Beekman: Tanya Latty and Madeleine Beekman, “Irrational Decision-Making in an Amoeboid Organism: Transitivity and Context-Dependent Preferences,” Proceedings of the Royal Society B: Biological Sciences 278, no. 1703 (Jan. 2011): 307–12.

  jays, honeybees, and hummingbirds: Susan C. Edwards and Stephen C. Pratt, “Rationality in Collective Decision-Making by Ant Colonies,” Proceedings of the Royal Society B: Biological Sciences 276, no. 1673 (2009): 3655–61.

  Psychologists Constantine Sedikides, Dan Ariely, and Nils Olsen: Constantine Sedikides, Dan Ariely, and Nils Olsen, “Contextual and Procedural Determinants of Partner Selection: Of Asymmetric Dominance and Prominence,” Social Cognition 17, no. 2 (1999): 118–39. But note also Shane Frederick, Leonard Lee, and Ernest Baskin, “The Limits of Attraction” (working paper), which argues that the evidence for the asymmetric domination effect in humans outside artificial lab scenarios is very weak.

  “among the very greatest improvements”: John Stuart Mill, On Liberty and Other Essays (Oxford: Oxford University Press, 1991), 310.

  the 2009 mayoral race in Burlington, Vermont: The vote totals here are all drawn from “Burlington Vermont IRV Mayor Election,” http://rangevoting.org/Burlington.html (accessed Jan. 15, 2014). See also University of Vermont political scientist Anthony Gierzynski’s assessment of the election, “Instant Runoff Voting,” www.uvm.edu/~vlrs/IRVassessment.pdf (accessed Jan. 15, 2014).

  “le mouton enragé”: Ian MacLean and Fiona Hewitt, eds., Condorcet: Foundations of Social Choice and Political Theory (Cheltenham, UK: Edward Elgar Publishing, 1994), 7.

  “I must act not by what I think reasonable”: From Condorcet’s “Essay on the Applications of Analysis to the Probability of Majority Decisions,” in Ian MacLean and Fiona Hewitt, Condorcet, 38.

  “mathematical charlatan”: Material on Condorcet, Jefferson, and Adams from MacLean and Hewitt, Condorcet, 64.

  made an extended visit to Voltaire’s house: The material about the relationship between Voltaire and Condorcet in this section is largely drawn from David Williams, “Signposts to the Secular City: The Voltaire-Condorcet Relationship,” in T. D. Hemming, Edward Freeman, and David Meakin, eds., The Secular City: Studies in the Enlightenment (Exeter, UK: University of Exeter Press, 1994), 120–33.

  Condorcet, like R. A. Fisher after him: Lorraine Daston, Classical Probability in the Enlightenment (Princeton, NJ: Princeton University Press, 1995), 99.

  “so beautiful that it was frightening”: recounted in a letter of Mme. Suard of June 3, 1775, quoted in Williams, “Signposts,” 128.

  Chapter 18: “Out of Nothing I Have Created a Strange New Universe”

  “You must not attempt this approach to parallels”: This quote, and much of the historical discussion of Bolyai’s work on non-Euclidean geometry, is drawn from Amir Alexander, Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics (Cambridge, MA: Harvard University Press, 2011), part 4.

  “To praise it would amount to praising myself”: Steven G. Krantz, An Episodic History of Mathematics (Washington, DC: Mathematical Association of America, 2010), 171.

  the U.S. Supreme Court said no: In Bush v. Gore, 531 U.S. 98 (2000).

  “Long live formalism”: Antonin Scalia, A Matter of Intepretation: Federal Courts and the Law (Princeton, NJ: Princeton University Press, 1997), 25.

  “The best-umpired game”: Quoted widely, e.g. in Paul Dickson, Baseball’s Greatest Quotations, rev. ed. (Glasgow: Collins, 2008), 298.

  Jeter knew it wasn’t a home run: To be fair, the question “What did Derek Jeter know and when did he know it?” has never been completely settled. In a 2011 interview with Cal Ripken, Jr., he conceded that the Yankees “caught a break” on the play but wasn’t willing to go so far as to say he should have been out. But he should have been out.

  “Most of the cases the Supreme Court agrees to decide”: from Richard A. Posner, “What’s the Biggest Flaw in the Opinions This Term?” Slate, June 21, 2013.

  what Congress must have meant: See, e.g., Scalia’s concurrence in Green v. Bock Laundry Machine Co., 490 U.S. 504 (1989).

  “When we are engaged in investigating”: From the translation of Hilbert’s speech by Mary Winston Newson, Bulletin of the American Mathematical Society, July 1902, 437–79.

  “tables, chairs, and beer mugs”: Reid, Hilbert, 57.

  “A careful reader”: Hilbert, “Über das unendliche,” Mathematische Annalen 95 (1926): 161–90; trans. Erna Putnam and Gerald J. Massey, “On the Infinite,” in Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics, 2d ed. (Cambridge, UK: Cambridge University Press, 1983).

  Terry Tao found a mistake: If you want to see what it looks like when serious mathematicians go toe-to-toe, you can watch the whole thing play out in real time in the comment section of the math blog The N-Category Café from September 27, 2011, “The Inconsistency of Arithmetic,” http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html (accessed Jan. 15, 2014).

  “The typical working mathematician”: Phillip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Houghton Mifflin, 1981), 321.

  Ramanujan was a prodigy from southern India: Robert Kanigel’s book The Man Who Knew Infinity (New York: Scribner, 1991) is a thorough popular account of Ramanujan’s life and work, if you want to know more.

  that was Hermann Minkowski: Reid, Hilbert, 7.

  Psychologists nowadays call it “grit”: See, e.g., the work of Angela Lee Duckworth.

  “It takes a thousand men”: From a letter Twain wrote on Mar. 17, 1903, to the young Helen Keller, available as “The Bulk of All Human Utterances Is Plagiarism,” Letters of Note, www.lettersofnote.com/2012/05/bulk-of-all-human-utterances-is.html (accessed Jan. 15, 2014).

  “The popular image”: Terry Tao, “Does One Have to Be a Genius to Do Maths?” http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths (accessed Jan. 15, 2014).

  As to the nature of the contradiction: The story and the quoted conversation come from “Kurt Gödel and the Institute,” Institute for Advanced Study, www.ias.edu/people/godel/institute.

  he refused to sign the 1914 Declaration to the Cultural World: Reid, Hilbert, 137.

  recounts the conversation: Constance Reid, Hilbert (Berlin: Springer-Verlag, 1970), 210.

  “We cannot usually avoid being presented with decisions of this kind”: From “An Election Between Three Candidates,” a section of Condorcet’s “Essay on the Applications of Analysis,” in MacLean and Hewitt, Condorcet.

  How to Be Right

  went his whole life without lifting a weapon in anger: Wald did fulfill his compulsory service in the Rumanian army,
though, so I can’t say for sure he didn’t.

  “Soonest Mended”: It appears in Ashbery’s 1966 book The Double Dream of Spring. You can read the poem online at www.poetryfoundation.org/poem/177260 (accessed Jan. 15, 2014).

  “Sitting on a fence”: “Sitting on a Fence” appears on the Housemartins’s debut record, London 0 Hull 4.

  I think of Silver as a kind of Kurt Cobain of probability: Some of this material is adapted from my review of Silver’s book The Signal and the Noise in the Boston Globe, Sept. 29, 2012.

  “On September 30, leading into the debates”: Josh Jordan, “Nate Silver’s Flawed Model,” National Review Online, Oct. 22, 2012, www.nationalreview.com/articles/331192/nate-silver-s-flawed-model-josh-jordan (accessed Jan. 15, 2014).

  “So should Mitt Romney win on Nov. 6”: Dylan Byers, “Nate Silver: One-Term Celebrity?” Politico, Oct. 29, 2012.

  states Silver considered potentially competitive: Nate Silver, “October 25: The State of the States,” New York Times, Oct. 26, 2012.

  “To believe something is to believe that it is true”: Willard Van Orman Quine, Quiddities: An Intermittently Philosophical Dictionary (Cambridge, MA: Harvard University Press, 1987), 21.

  Obama had a 67% chance: These aren’t Silver’s actual numbers, which aren’t archived as far as I can tell, just numbers made up to illustrate the kind of predictions he was making before the election.

  Sometimes the people speak and they say “I dunno”: The discussion of close elections is adapted from my article “To Resolve Wisconsin’s State Supreme Court Election, Flip a Coin,” Washington Post, Apr. 11, 2011.

  “He knew little out of his way”: From The Autobiography of Benjamin Franklin (New York: Collier, 1909), www.gutenberg.org/cache/epub/148/pg148.html (accessed Jan. 15, 2014).

  their reasoning modules frazzle and halt: See, e.g., “I, Mudd,” Star Trek, air date Nov. 3, 1967.

  “The test of a first-rate intelligence”: F. Scott Fitzgerald, “The Crack-Up,” Esquire, Feb. 1936.

  it’s said of the topologist R. H. Bing: For instance, in George G. Szpiro, Poincaré’s Prize: The Hundred-Year Quest to Solve One of Math’s Greatest Puzzles (New York: Dutton, 2007).

  “I can’t go on, I’ll go on”: Samuel Beckett, The Unnameable (New York: Grove Press, 1958).

  Determined to record and neutralize: My take on DFW’s language is taken from an article I published in Slate on Sept. 18, 2008, “Finite Jest: Editors and Writers Remember David Foster Wallace,” www.slate.com/articles/arts/culturebox/2008/09/finite_jest_2.html.

  “and you go the way of Hippasos”: Samuel Beckett, Murphy (London: Routledge, 1938).

  INDEX

  The page numbers in this index refer to the printed version of this book. The link provided will take you to the beginning of that print page. You may need to scroll forward from that location to find the corresponding reference on your e-reader.

  ABC News, 50

  Abel, Niels Henrik, 48

  Abraham, 89–90

  Adams, John, 388

  additivity of expected value, 212, 214–22

  Affordable Care Act, 368

  Against the Day (Pynchon), 324

  Ahmadinejad, Mahmoud, 173

  airport wait times versus missed flights, 233–36

  Albrecht, Spike, 125, 126, 129

  Alcmaeon of Croton, 262

  Alexander the Great, 17

  algebra, 104–10, 336

  algorithms, 56–58

  asteroid trajectory, forecasting, 164

  Target’s pregnancy predicting, 163, 166, 170

  terrorist-finding, 166–71

  weather forecasting, 164–65

  for solving equations in whole numbers, 405

  Alhazen. See Ibn al-Haytham, ‘Ali al-Hasan

  Allen, Scott, 227

  alliteration, statistical analysis of, 123–25

  Amar, Akhil, 373, 375

  Amar, Vikram, 373, 375

  Amgen, 150

  annuities, pricing of, 199–200

  Appolonius of Perga, 319–20, 322

  a priori probabilities, 175

  Arbuthnot, John, 115, 116

  Archimedes, 31, 32, 34–38, 40, 221

  area, of circle, 31–39

  “An Argument for Divine Providence, Taken from the Constant Regularity Observed in the Births of Both Sexes” (Arbuthnot), 115

  argument by design, 185–91

  Ariely, Dan, 382

  Aristotle, 24, 99, 131

  Arrow, Kenneth, 419

  Ashbery, John, 424–25

  Association for Psychological Science, 162

  asteroid trajectory, forecasting, 164

  asymmetric domination effect, 382

  Atkins, Daryl Renard, 370, 372

  Atkins v. Virginia, 370–76, 403

  atrocities

  as partially ordered set, 75–76

  proportional comparison of, 62–65, 74–75

  Attali, Yigal, 129

  “Axiomatic Thought” (Hilbert), 416

  Bakan, David, 117

  Baltimore stockbroker, 95–99, 152

  Banach-Tarski paradox, 391

  Barbier, Jospeh-Émile, 217, 218, 222, 223

  Bar-Natan, Dror, 99–100, 101

  Baseball

  formalism in, 403

  regression to the mean in, 304–6

  basketball

  “hot hand,” statistical analysis of, 125–28

  NBA shooting percentages, and ranking systems, 68–69

  Bayesian inference, 171–91

  argument by design and, 186–91

  mechanics of, 174–78

  Bayes theorem, 178

  Beal, Andrew, 143

  Beaudrot, Nicholas, 236

  Beautiful Mind, A (movie), 223–24

  Beber, Bernd, 173

  Beckett, Samuel, 434–36

  Beekman, Madeleine, 376, 381

  Beethoven, Ludwig van, 328

  Bell, 272, 276

  bell curve, 68–74

  Benitez, Armando, 402–3

  Bennett, Craig, 102–4

  Bennett, Jim, 166

  Berkeley, George, 40, 46

  Berkson, Jospeh, 136, 357–59

  Berkson’s fallacy, 357–62

  Bernoulli, Daniel, 243, 244–46, 247, 248, 252, 271

  Bernoulli, Jakob, 68

  Bernoulli, Nicolas, 115–16, 243

  Bertillon, Alphonse, 325–26

  bertillonage, 325–29

  Bertrand, Joseph, 223

  Better Angels of Our Nature, The (Pinker), 64

  Bible Code, The (Drosnin), 93–94

  Bible codes, 89–95, 99–101, 111, 114–15

  binary variables, and correlation, 347–49

  Bing, R. H., 433

  bits, 269

  bivariate normal distribution, 319

  Blumenthal, Otto, 418

  Bolyai, Farkas, 395

  Bolyai, János, 395–96, 399, 434, 436

  Book of Isaiah, 93

  Borda, Jean-Charles de, 379–80, 390

  Bose-Chaudhuri-Hocquenghem codes, 276

  Boston Globe, 227, 228, 231

  Bostrom, Nick, 189

  bounded gaps conjecture, 138, 140–42, 144

  brain cancer incidence, 65

  British Medical Journal, 309

  Brooks, David, 341

  Broom of the System, The (Wallace), 435

  Brown, Derren, 97

  Brunelleschi, Filippo, 262, 264, 398

  Buchanan, Pat, 400

  Buffon, Georges-Louis LeClerc, Comte de, 214–17, 222, 223, 247–48

  Buffon’s needle problem, 216–22

  Buffon’s noodle p
roblem, 222

  bullet hole problem, 5–8, 9–10

  Burlington, Vermont 2009 mayoral election, 384–87, 391–92

  Burney, Leroy E., 353, 354

  Bush, George H. W., 369, 374

  Bush, George W., 322, 341, 378–80, 383–84, 400–401

  Bush v. Gore, 404

  business performance, and regression to the mean, 295–98, 302

  Byers, Dylan, 427

  Byman, Daniel, 62

  calculus, 39–49, 56

  Candès, Emmanuel, 330

  Cantor, Georg, 271, 410

  capacity of a communication channel, 272

  capital punishment, 370–76

  Caplan, Bryan, 366

  Caplan, Gerald, 62–63

  calculus, 12, 39–41, 47

  Caramello, Olivia, 399

  Cardano, Girolamo, 67–68, 78

  Cash Winfall, 206–13, 224–32, 253, 256–61, 258–60

  cat in the hat problem, 184–85

  Cato Institute, 21–23

  Cauchy, Augustin-Louis, 47–49

  causal relationships, and correlation, 335–36, 349–62

  Chabris, Christopher, 150

  Chandler, Tyson, 69

  Chao, Lu, 431–32

  chaos, 164–65

  checksum, 276

  Cheney, Dick, 25

  Churchill, Winston, 265

  cigarette smoking/lung cancer correlation, 350–55

  circle, area of, 31–39

  circumscribed square, 36

  “Citizenship in a Republic” (Roosevelt), 422–23

  Clausewitz, Carl von, 13

  cleanest man in school, 188

  clinical trials. See drug efficacy studies

  Clinton, Bill, 77, 369, 374

  CNN, 93

  Cobain, Kurt, 426

  code words, 273

  Cohn, Henry, 281

  coin flips, 65–68, 71–74

  Coleman, Norm, 431

  combinatorial explosion, 261

  common sense, 9–14

  compact discs, 276

  compression, 330

  conditional expectation, 314

  conditional probabilities, 170

  Condorcet, Marie-Jean-Antoine-Nicolas de Caritat, Marquis de, 387–93, 400–401, 418–20, 424, 436