How Not to Be Wrong

Jordan Ellenberg argues that mathematics is not a set of arcane procedures confined to classrooms and specialists but an extension of common sense, a disciplined way of thinking that touches politics, medicine, commerce, and virtually every domain of daily life. Organized into five thematic parts, the book moves through real-world problems and historical episodes to show how mathematical reasoning protects us from errors that seem obvious only in hindsight.

Ellenberg opens with the story of Abraham Wald, a Jewish mathematician who fled Nazi-occupied Austria and joined a classified group of statisticians in wartime New York, the Statistical Research Group. When the military proposed armoring planes where returning aircraft showed the most bullet holes, Wald recognized what the officers missed: The holes marked where planes could absorb damage and survive. The missing holes, on the engines, indicated where hits brought planes down before they could return. The officers had mistakenly treated surviving planes as representative of all planes, a mistake Ellenberg identifies as survivorship bias. He connects this same error to mutual fund performance, where funds that collapse during a measurement period vanish from the record, inflating the apparent returns of the survivors.

The first section addresses linearity, the widespread but often false assumption that relationships between variables follow a straight line. Ellenberg uses the Laffer curve, economist Arthur Laffer's illustration that both zero and total taxation produce zero government revenue, to show that the relationship between policy and outcomes is typically nonlinear. If the curve has a hump in the middle, the optimal policy depends on where a country currently stands, and two nations might rationally move in opposite directions. He introduces the foundational idea of calculus, that any smooth curve viewed at a sufficiently small scale resembles a straight line, and traces the concept from Archimedes through Isaac Newton. He then shows how assuming linearity leads to absurd results: A 2008 paper in the journal Obesity projected all Americans would be overweight by 2048, and the Romney presidential campaign claimed women accounted for 92.3% of jobs lost under Obama by exploiting the near-cancellation of male job losses and gains.

The second part turns to inference. Ellenberg explains the null hypothesis significance test developed by R. A. Fisher: One assumes the effect under study does not exist, computes the probability (the p-value) of obtaining results as extreme as those observed, and rejects the null hypothesis if the p-value falls below a threshold, typically 0.05. The Witztum-Rips-Rosenberg paper, published in Statistical Science in 1994, claimed that equidistant letter sequences in Genesis encoded the names of medieval rabbis at rates exceeding chance. Mathematicians later showed that small changes in spelling could produce equally impressive results in War and Peace, demonstrating how flexibility in experimental design generates apparently miraculous findings. Neuroscientist Craig Bennett's satirical study found statistically significant brain activity in a dead salmon by failing to correct for multiple comparisons, the problem of testing many patterns at once, which inflates false positives.

Drawing on biomedical researcher John Ioannidis, Ellenberg explains why many published findings may be false: When researchers test thousands of hypotheses, as in genome-wide association studies that scan many genetic variants for links to disease, the false positive rate produces far more spurious results than genuine ones. Problems multiply through the file drawer effect (negative results go unpublished), p-hacking (adjusting analyses until they cross the significance threshold), and the winner's curse (significant effects in small studies are almost certainly overestimated). He discusses the philosophical divide between Fisher, who saw significance as a clue pointing toward truth, and statisticians Jerzy Neyman and his collaborator Egon Pearson, who treated it as a decision rule analogous to a criminal verdict.

Ellenberg then introduces Bayesian inference, which incorporates prior beliefs alongside new evidence. The same data can lead to different conclusions depending on one's starting assumptions, which explains why we rightly treat a promising drug trial and a test of plastic-Stonehenge healing differently even when both produce identical p-values. He extends the framework to the argument by design for God's existence, using it to show that quantitative reasoning reaches its limits on ultimate metaphysical questions.

The third part develops expected value, the average outcome of a decision repeated many times. Ellenberg shows that a standard Powerball ticket is almost always a losing proposition but that the Massachusetts Cash WinFall game offered positive expected value on "roll-down" days when the unclaimed jackpot cascaded into lower prize tiers. MIT student James Harvey and two other syndicates independently exploited this loophole. Ellenberg uses the principle of additivity of expected value to solve Buffon's needle problem, showing that the probability a dropped needle crosses a line on a ruled floor is 2/π. He extends the discussion to the St. Petersburg paradox, a game with infinite expected dollar value that no one would pay much to play. Mathematician Daniel Bernoulli resolved the paradox by showing that the utility of money is nonlinear: Each additional dollar matters less than the last. RAND analyst Daniel Ellsberg's paradox demonstrated that people distinguish between quantifiable risk and genuine uncertainty in ways the standard expected utility theory, which holds that rational decisions maximize a weighted average of outcomes, cannot accommodate.

The section connects the lottery problem to coding theory, the study of encoding messages for reliable transmission. Ellenberg shows that the Hamming code, an error-correcting scheme developed by Richard Hamming, shares the same mathematical structure as the Fano plane, a minimal geometry of seven points and seven lines, and the optimal ticket selection for a small lottery. These connections illustrate a recurring theme: Seemingly unrelated problems often share hidden mathematical structures.

The fourth part addresses regression to the mean, the tendency for extreme observations to be followed by less extreme ones. Ellenberg traces the concept to statistician Francis Galton's study of hereditary height: Tall parents tend to have children who are tall but not quite as tall, because extreme height reflects both genetic predisposition and favorable chance, and the chance component is unlikely to recur. Scholar Horace Secrist's 1933 book, The Triumph of Mediocrity in Business, documented the regression of top-performing companies toward average performance and concluded that competition drives excellence toward mediocrity. Statistician Harold Hotelling's review demonstrated that the phenomenon was a mathematical inevitability, not evidence of competitive forces. Regression to the mean also explains the apparent decline of league-leading home run hitters and the failure of the Scared Straight program, which randomized trials showed actually increased antisocial behavior.

This section develops correlation, Galton's measure of the association between two variables. Ellenberg demonstrates that correlation is not transitive: Rich individuals are more likely to vote Republican even though rich states tend to vote Democratic. He applies this non-transitivity to medicine, explaining why niacin raises HDL cholesterol without reducing heart attacks. He introduces Berkson's fallacy, by which spurious correlations arise from a common effect rather than a common cause: Hospital studies can create false associations between diseases because patients are selected for being sick.

The fifth part examines the limits of mathematical formalism, the approach that treats mathematical truth as purely a matter of deriving conclusions from stated axioms, in democratic decision-making and in the foundations of mathematics itself. Ellenberg demonstrates that majority preferences can be internally contradictory: Americans want to cut spending but oppose cutting any specific program, not because voters are irrational but because aggregating individually consistent views produces incoherent collective preferences. He presents the Condorcet paradox, named after the Marquis de Condorcet, the Enlightenment mathematician who discovered it: With three or more candidates, majority preferences can cycle, making it impossible to identify a clear winner. Economist Kenneth Arrow's 1951 impossibility theorem proved that no voting system satisfying a small set of reasonable axioms can avoid such paradoxes.

Ellenberg uses the parallel postulate of Euclidean geometry—the axiom stating that through any point not on a given line, exactly one line can be drawn parallel to it—to illustrate formalism. Mathematician János Bolyai discovered non-Euclidean geometry, an alternative system in which this axiom does not hold, by showing that the parallel postulate cannot be derived from Euclid's other axioms. Formalist mathematician David Hilbert sought to rebuild all of mathematics on explicitly consistent axioms, but logician Kurt Gödel's 1931 incompleteness theorem showed that no such consistency proof is possible within arithmetic. Despite this, Ellenberg argues, formalism persists as practical methodology: Mathematicians use intuition to discover results but rely on formal proof to verify them.

Ellenberg concludes by arguing that principled uncertainty is itself a form of action. He cites statistician and election forecaster Nate Silver's 2012 work as exemplary: Silver reported probability distributions rather than predictions, drawing criticism from pundits but proving more accurate than any of them. The book's final message is that mathematical thinking, which involves recognizing nonlinearity, respecting improbable events, weighing possible futures, and distinguishing individual from collective rationality, is something everyone already does, and doing it more deliberately is the surest path to not being wrong.