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

by Ellenberg, Jordan

  I thank my son and daughter for being patient through the many working weekends the book has made necessary, and my son in particular for drawing one of the pictures. And most of all, Tanya Schlam, both first and final reader of everything you’ve seen here, and the person whose support and love made it possible even to conceive this project. She has helped me understand, even more than mathematics has, how to be right.

  NOTES

  When Am I Going to Use This?

  Abraham Wald was born in 1902: Biographical material about Abraham Wald is drawn from Oscar Morgenstern, “Abraham Wald, 1902–1950,” Econometrica 19, no. 4 (Oct. 1951): 361–67.

  The Statistical Research Group (SRG): Historical material about the SRG is largely drawn from W. Allen Wallis, “The Statistical Research Group, 1942–1945,” Journal of the American Statistical Association 75, no. 370 (June 1980): 320–30.

  “When we made recommendations”: Ibid., 322.

  “the most extraordinary group”: Ibid., 322.

  the joke around SRG: Ibid., 329.

  So here’s the question: I learned about Wald and the missing bullet holes from Howard Wainer’s book Uneducated Guesses: Using Evidence to Uncover Misguided Education Policies (Princeton, NJ: Princeton University Press, 2011) where he applies Wald’s insights to the similarly complicated and partial statistics obtained in education studies.

  through the wars in Korea and Vietnam: Marc Mangel and Francisco J. Samaniego, “Abraham Wald’s Work on Aircraft Survivability,” Journal of the American Statistical Association 79, no. 386 (June 1984): 259–67.

  “all of the most abstract sort”: Jacob Wolfowitz, “Abraham Wald, 1902–1950,” Annals of Mathematical Statistics 23, no. 1 (Mar. 1952): 1–13.

  A 2006 study by Savant Capital: Amy L. Barrett and Brent R. Brodeski, “Survivor Bias and Improper Measurement: How the Mutual Fund Industry Inflates Actively Managed Fund Performance,” www.savantcapital.com/uploadedFiles/Savant_CMS_Website/Press_Coverage/Press_Releases/Older_releases/sbiasstudy[1].pdf (accessed Jan. 13, 2014).

  a comprehensive 2011 study in the Review of Finance: Martin Rohleder, Hendrik Scholz, and Marco Wilkens, “Survivorship Bias and Mutual Fund Performance: Relevance, Significance, and Methodical Differences,” Review of Finance 15 (2011): 441–74; see table. We have converted monthly excess return to annual excess return, so the numbers in the text don’t match those in the table.

  Wald’s actual report: Abraham Wald, A Method of Estimating Plane Vulnerability Based on Damage of Survivors (Alexandria, VA: Center for Naval Analyses, repr., CRC 432, July 1980).

  books of its own: For the Riemann Hypothesis, I like John Derbyshire’s Prime Obsession and Marcus du Sautoy’s The Music of the Primes. For Gödel’s Theorem, there’s of course Douglas Hofstadter’s Gödel, Escher, Bach, which is, to be fair, only tangentially about the theorem as one mantra in its meditation on self-reference in art, music, and logic.

  Chapter 1: Less like Sweden

  a blog entry with the provocative title: Daniel J. Mitchell, “Why Is Obama Trying to Make America More Like Sweden when Swedes Are Trying to Be Less Like Sweden?” Cato Institute, Mar. 16, 2010, www.cato.org/blog/why-obama-trying-make-america-more-sweden-when-swedes-are-trying-be-less-sweden (accessed Jan. 13, 2014).

  “Est modus in rebus”: Horace, Satires 1.1.106, trans. Basil Dufallo, in “Satis/Satura: Reconsidering the ‘Programmatic Intent’ of Horace’s Satires 1.1,” Classical World 93 (2000): 579–90.

  the government does take in some amount of revenue: Laffer was always very clear on the point that the Laffer curve was not his invention; Keynes had understood and written about the idea very clearly, and the basic idea goes back (at least) as far as the fourteenth-century historian Ibn Khaldun.

  “Thomas Edison was a nut”: Jonathan Chait, “Prophet Motive,” New Republic, Mar. 31, 1997.

  “and he can talk about it for six months”: Hal R. Varian, “What Use Is Economic Theory?” (1989), http://people.ischool.berkeley.edu/~hal/Papers/theory.pdf (accessed Jan. 13, 2014).

  “I came into the Big Money”: David Stockman, The Triumph of Politics: How the Reagan Revolution Failed (New York: Harper & Row, 1986), 10.

  “Subsequent history failed to confirm Laffer’s conjecture”: N. Gregory Mankiw, Principles of Microeconomics, vol. 1 (Amsterdam: Elsevier, 1998), 166.

  his acid assessment of supply-side theory: Martin Gardner, “The Laffer Curve,” The Night Is Large: Collected Essays, 1938-1995 (New York: St. Martin’s, 1996), 127–39.

  during congressional testimony: In 1978, during consideration of the Kemp-Roth tax-cut bill.

  Chapter 2: Straight Locally, Curved Globally

  one of the very few ancient Greeks to wear pants: Christoph Riedweg, Pythagoras: His Life, Teaching, and Influence (Ithaca, NY: Cornell University Press, 2005), 2.

  “the ghosts of departed quantities”: George Berkeley, The Analyst: A Discourse Addressed to an Infidel Mathematician (1734), ed. David R. Wilkins, www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analyst.pdf (accessed Jan. 13, 2014).

  Most people, if you press them: David O. Tall and Rolph L. E. Schwarzenberger, “Conflicts in the Learning of Real Numbers and Limits,” Mathematics Teaching 82 (1978): 44–49.

  2-adic numbers: In Cauchy’s theory, a series converging to a limit x means that when you sum more and more terms, the total gets closer and closer to x. This requires that we have in mind an idea of what it means for two numbers to be “close” to each other. It turns out that the familiar notion of closeness is not the only one! In the 2-adic world, two numbers are said to be close to each other when their difference is a multiple of a large power of 2. When we say the series 1 + 2 + 4 + 8 + 16 + . . . converges to −1, we are saying that the partial sums 1, 3, 7, 15, 31 . . . are getting closer and closer to −1. With the usual meaning of “close,” that’s not true; but using 2-adic closeness, it’s a different story. The numbers 31 and −1 differ by 32, which is 25, a pretty small 2-adic number. Sum a few more terms and you get 511, which differs from –1 by only 512, smaller (2-adically) still. Much of the math you know—calculus, logarithms and exponentials, geometry—has a 2-adic analogue (and indeed a p-adic analogue for any p), and the interaction between all these different notions of closeness is a crazy and glorious story of its own.

  including the Italian mathematician/priest Guido Grandi: Material on Grandi and his series is largely drawn from Morris Kline, “Euler and Infinite Series,” Mathematics Magazine 56, no. 5 (Nov. 1983): 307–14.

  Cauchy was interested in the truth: The story of Cauchy’s calculus class is taken from Duel at Dawn, Amir Alexander’s immensely interesting historical study of the interaction between math and culture at the beginning of the nineteenth century. See also Michael J. Barany, “Stuck in the Middle: Cauchy’s Intermediate Value Theorem and the History of Analytic Rigor,” Notices of the American Mathematical Society 60, no. 10 (Nov. 2013): 1334–38, for a somewhat contrary point of view concerning the modernness of Cauchy’s approach.

  Chapter 3: Everyone Is Obese

  a paper in the journal Obesity: Youfa Wang et al., “Will All Americans Become Overweight or Obese? Estimating the Progression and Cost of the US Obesity Epidemic,” Obesity 16, no. 10 (Oct. 2008): 2323–30.

  “obesity apocalypse”: abcnews.go.com/Health/Fitness/story?id=5499878&page=1.

  “We’re Getting Fatter”: Long Beach Press-Telegram, Aug. 17, 2008.

  We’re not all going to be overweight: My discussion of the Wang obesity study largely agrees with that in Carl Bialik’s article “Obesity Study Looks Thin” (Wall Street Journal, Aug. 15, 2008), which I learned about after writing this chapter.

  the North Carolina Career Resource Network: The figures here are from www.soicc.state.nc.us/soicc/planning/c2c.htm, which has since been taken down.

  had already begun to slow: Katherine M. Flegal et al., “Prevalence of Obesity and Trends
in the Distribution of Body Mass Index Among US Adults, 1999–2010,” Journal of the American Medical Association 307, no. 5 (Feb. 1, 2012), 491–97.

  Chapter 4: How Much Is That in Dead Americans?

  “The Israeli military reports”: Daniel Byman, “Do Targeted Killings Work?” Foreign Affairs 85, no. 2 (Mar.–Apr. 2006), 95.

  “the equivalent, on a proportional basis”: “Expressing Solidarity with Israel in the Fight Against Terrorism,” H. R. Res. 280, 107th Congress (2001).

  Newt Gingrich: Some of the material in this chapter is adapted from my article “Proportionate Response,” Slate, July 24, 2006.

  “Remember that when Israel loses eight people”: From Meet the Press, July 16, 2006, transcript at www.nbcnews.com/id/13839698/page/2/#.Uf_Gc2TEo9E (accessed Jan. 13, 2014).

  “When Israel killed 1,400 Palestinians in Gaza”: Ahmed Moor, “What Israel Wants from the Palestinians, It Takes,” Los Angeles Times, Sept. 17, 2010.

  “Some 45,000 Nicaraguans”: Gerald Caplan, “We Must Give Nicaragua More Aid,” Toronto Star, May 8, 1988.

  “equivalent to 27 million Americans”: David K. Shipler, “Robert McNamara and the Ghosts of Vietnam,” New York Times Magazine, Aug. 10, 1997, pp. 30–35.

  they have the most people: The brain cancer data is all from “State Cancer Profiles,” National Cancer Institute, http://statecancerprofiles.cancer.gov/cgi-bin/deathrates/deathrates.pl?00&076&00&2&001&1&1&1 (accessed Jan. 13, 2014).

  much more or much less likely you’ll get brain cancer: The example of brain cancer rates owes much to a similar treatment of county-by-county kidney cancer statistics in Howard Wainer’s book Picturing the Uncertain World (Princeton, NJ: Princeton University Press, 2009), which develops the idea much more thoroughly than I do here.

  10,000 times in all: John E. Kerrich, “Random Remarks,” American Statistician 15, no. 3 (June 1961), 16–20.

  Who wins this kind of contest?: The scores for 1999 are taken from “A Report Card for the ABCs of Public Education Volume I: 1998-1999 Growth and Performance of Public Schools in North Carolina—25 Most Improved K-8 Schools,” www.ncpublicschools.org/abc_results/results_99/99ABCsTop25.pdf (accessed Jan. 13, 2014).

  shooting percentage is as much a function: Kirk Goldsberry, “Extra Points: A New Way to Understand the NBA’s Best Scorers,” Grantland, Oct. 9, 2013, www.grantland.com/story/_/id/9795591/kirk-goldsberry-introduces-new-way-understand-nba-best-scorers (accessed Jan. 13, 2014), suggests one way of going beyond shooting percentage to develop more informative measures of offensive performance.

  A study by Thomas Kane and Douglas Staiger: Thomas J. Kane and Douglas O. Staiger, “The Promise and Pitfalls of Using Imprecise School Accountability Measures,” Journal of Economic Perspectives 16, no. 4 (Fall 2002), 91–114.

  I’ll spare you here: But see Kenneth G. Manton et al., “Empirical Bayes Procedures for Stabilizing Maps of U.S. Cancer Mortality Rates,” Journal of the American Statistical Association 84, no. 407 (Sept. 1989): 637–50; and Andrew Gelman and Phillip N. Price, “All Maps of Parameter Estimates Are Misleading,” Statistics in Medicine 18, no. 23 (1999): 3221–34) if you want the no-holds-barred technical treatment.

  the gendarme’s hat: Stephen M. Stigler, Statistics on the Table: The History of Statistical Concepts and Methods (Cambridge, MA: Harvard University Press, 1999), 95.

  decipherable formulae: See, e.g., Ian Hacking, The Emergence of Probability: A Philosophical Study of Early Ideas About Probability, Induction, and Statistical Inference, 2d ed. (Cambridge, UK: Cambridge University Press, 2006), ch. 18.

  King Leopold’s war in the Congo: White’s figures here are taken from Matthew White, “30 Worst Atrocities of the 20th Century,” http://users.erols.com/mwhite28/atrox.htm (accessed Jan. 13, 2014).

  Chapter 5: More Pie than Plate

  A recent working paper by economists Michael Spence and Sandile Hlatshwayo: A. Michael Spence and Sandile Hlatshwayo, “The Evolving Structure of the American Economy and the Employment Challenge,” Council on Foreign Relations, Mar. 2011, www.cfr.org/industrial-policy/evolving-structure-american-economy-employment-challenge/p24366 (accessed Jan. 13, 2014).

  From The Economist: “Move Over,” Economist, July 7, 2012.

  Bill Clinton’s latest book: William J. Clinton, Back to Work: Why We Need Smart Government for a Strong Economy (New York: Random House, 2011), 167.

  ficta, or fake: Jacqueline A. Stedall, From Cardano’s Great Art to Lagrange’s Reflections: Filling a Gap in the History of Algebra (Zurich: European Mathematical Society, 2011), 14.

  “over 50 percent of U.S. job growth in June came from our state”: Milwaukee Journal Sentinel, PolitiFact, www.politifact.com/wisconsin/statements/2011/jul/28/republican-party-wisconsin/wisconsin-republican-party-says-more-than-half-nat (accessed Jan. 13, 2014).

  “Something we are doing here must be working”: WTMJ News, Milwaukee, “Sensenbrenner, Voters Take Part in Contentious Town Hall Meeting over Federal Debt,” www.todaystmj4.com/news/local/126122793.html (accessed Jan. 13, 2014).

  more than thirteen thousand in the same month: All the job data here comes from the June 2011 Regional and State Employment and Unemployment (Monthly) News Release by the Bureau of Labor Statistics, July 22, 2011, www.bls.gov/news.release/archives/laus_07222011.htm.

  New York Times op-ed: Steven Rattner, “The Rich Get Even Richer,” New York Times, Mar. 26, 2012, A27.

  which they’ve helpfully put online: elsa.berkeley.edu/~saez/TabFig2010.xls (accessed Jan. 13, 2014).

  released a statement asserting: Mitt Romney, “Women and the Obama Economy,” Apr. 10, 2012, available at www.scribd.com/doc/88740691/Women-And-The-Obama-Economy-Infographic.

  one month later, in February 2009: The computations and argument here are drawn from Glenn Kessler, “Are Obama’s Job Policies Hurting Women?” Washington Post, Apr. 10, 2012.

  “true but false”: Ibid.

  Chapter 6: The Baltimore Stockbroker and the Bible Code

  “After Abraham was weaned”: Maimonides, Laws of Idolatry 1.2, from Isadore Twersky, A Maimonides Reader (New York: Behrman House, Inc., 1972), 73.

  respite for Slovakia’s Jews: Yehuda Bauer, Jews for Sale? Nazi-Jewish Negotiations, 1933–1945 (New Haven: Yale University Press, 1996), 74–90.

  Witztum and his colleagues: Doron Witztum, Eliyahu Rips, and Yoav Rosenberg, “Equidistant Letter Sequences in the Book of Genesis,” Statistical Science 9, no. 3 (1994): 429–38.

  “Our referees were baffled”: Robert E. Kass, “In This Issue,” Statistical Science 9, no. 3 (1994): 305–6.

  “they have disgraced mathematics”: Shlomo Sternberg, “Comments on The Bible Code,” Notices of the American Mathematical Society 44, no. 8 (Sept. 1997): 938–39.

  a practice called incubation: Alan Palmiter and Ahmed Taha, “Star Creation: The Incubation of Mutual Funds,” Vanderbilt Law Review 62 (2009): 1485–1534. Palmiter and Taha explicitly draw the analogy between the Baltimore stockbroker and fund incubation.

  the same returns as the median fund: Ibid., 1503.

  a much less miraculous one in a thousand: Leonard A. Stefanski, “The North Carolina Lottery Coincidence,” American Statistician 62, no. 2 (2008): 130–34.

  “what is improbable is probable”: Aristotle, Rhetoric 2.24, trans. W. Rhys Roberts, classics.mit.edu/Aristotle/rhetoric.mb.txt (accessed Jan. 14, 2014).

  “the ‘one chance in a million’ will undoubtedly occur”: Ronald A. Fisher, The Design of Experiments (Edinburgh: Oliver & Boyd, 1935), 13–14.

  McKay and Bar-Natan found: Brendan McKay and Dror Bar-Natan, “Equidistant Letter Sequences in Tolstoy’s War and Peace,” cs.anu.edu.au/~bdm/dilugim/WNP/main.pdf (accessed Jan. 14, 2014).

  “equally appalling”: Brendan McKay, Dror Bar-Natan, Maya Bar-Hillel, and Gil Kalai, “Solving the Bible Code Puzzle,” Statistical Science 14, no. 2 (1999): 150–73, section 6.
/>   In a later paper: Ibid.

  a full-page ad: New York Times, Dec. 8, 2010, A27.

  Witztum, Rips, and Rosenberg insist: See, e.g., Witztum’s article “Of Science and Parody: A Complete Refutation of MBBK’s Central Claim,” www.torahcode.co.il/english/paro_hb.htm (accessed Jan. 14, 2014).

  Chapter 7: Dead Fish Don’t Read Minds

  “Neural correlates”: Craig M. Bennett et al., “Neural Correlates of Interspecies Perspective Taking in the Post-Mortem Atlantic Salmon: An Argument for Proper Multiple Comparisons Correction,” Journal of Serendipitous and Unexpected Results 1 (2010): 1–5.

  “One mature Atlantic Salmon”: Ibid., 2

  The joke, like all jokes: Gershon Legman, Rationale of the Dirty Joke: An Analysis of Sexual Humor (New York: Grove, 1968; repr. Simon & Schuster, 2006).

  are said to understand it: See, e.g., Stanislas Dehaene, The Number Sense: How the Mind Creates Mathematics (New York: Oxford University Press, 1997)

  in cryptic rhymed verse: Richard W. Feldmann, “The Cardano-Tartaglia Dispute,” Mathematics Teacher 54, no. 3 (1961): 160–63.

  and part-time mathematician: Material on Arbuthnot drawn from chapter 18 of Ian Hacking, The Emergence of Probability (New York: Cambridge University Press, 1975) and chapter 6 of Stephen M. Stigler, The History of Statistics (Cambridge, MA: Harvard University Press/Belknap Press, 1986).

  to look like the one we have: See Elliot Sober’s Evidence and Evolution: The Logic Behind the Science (New York: Cambridge University Press, 2008) for a thorough discussion of the many, many strands, both classical and contemporary, of this “argument by design.”