Mathematics

A fast-food chain once tried to compete with McDonald’s quarter-pounder by introducing a third-pound hamburger—only for it to flop when consumers thought a third pound was less than a quarter pound because three is less than four. Separately, a rash of suicides by teenagers who played Dungeons and Dragons caused a panic in parents and the media. They thought D&D was causing teenage suicides—when in fact teenage D&D players died by suicide at a much lower rate than the national average. Errors of this type can be found from antiquity to the present, from the Peloponnesian War to the COVID-19 pandemic. How and why do we keep falling into these traps?

James C. Zimring argues that many of the mistakes that the human mind consistently makes boil down to misperceiving fractions. We see slews of statistics that are essentially fractions, such as percentages, probabilities, frequencies, and rates, and we tend to misinterpret them. Sometimes bad actors manipulate us by cherry-picking data or distorting how information is presented; other times, sloppy communicators inadvertently mislead us. In many cases, we fool ourselves and have only our own minds to blame. Zimring also explores the counterintuitive reason that these flaws might benefit us, demonstrating that individual error can be highly advantageous to problem solving by groups. Blending key scientific research in cognitive psychology with accessible real-life examples, Partial Truths helps readers spot the fallacies lurking in everyday information, from politics to the criminal justice system, from religion to science, from business strategies to New Age culture.

A fast-food chain once tried to compete with McDonald’s quarter-pounder by introducing a third-pound hamburger—only for it to flop when consumers thought a third pound was less than a quarter pound because three is less than four. Separately, a rash of suicides by teenagers who played Dungeons and Dragons caused a panic in parents and the media. They thought D&D was causing teenage suicides—when in fact teenage D&D players died by suicide at a much lower rate than the national average. Errors of this type can be found from antiquity to the present, from the Peloponnesian War to the COVID-19 pandemic. How and why do we keep falling into these traps?

James C. Zimring argues that many of the mistakes that the human mind consistently makes boil down to misperceiving fractions. We see slews of statistics that are essentially fractions, such as percentages, probabilities, frequencies, and rates, and we tend to misinterpret them. Sometimes bad actors manipulate us by cherry-picking data or distorting how information is presented; other times, sloppy communicators inadvertently mislead us. In many cases, we fool ourselves and have only our own minds to blame. Zimring also explores the counterintuitive reason that these flaws might benefit us, demonstrating that individual error can be highly advantageous to problem solving by groups. Blending key scientific research in cognitive psychology with accessible real-life examples, Partial Truths helps readers spot the fallacies lurking in everyday information, from politics to the criminal justice system, from religion to science, from business strategies to New Age culture.

Why does it always seem like the elevator is going down when you need to go up? Is it really true that 0.99999 . . . with an infinite number of 9s after the decimal point, is equal to 1? What do tea leaves and river erosion have in common, per Albert Einstein? Does seeing a bed of red flowers help prove that all ravens are black? Can we make sense of a phrase like “this statement is unprovable”?

Exploring these questions and many more, George G. Szpiro guides readers through the puzzling world of paradoxes, from Socratic dialogues to the Monty Hall problem. Perplexing Paradoxes presents sixty counterintuitive conundrums drawn from diverse areas of thought—not only mathematics, statistics, logic, and philosophy but also social science, physics, politics, and religion. Szpiro offers a brisk history of each paradox, unpacks its inner workings, and considers where one might encounter it in daily life. Ultimately, he argues, paradoxes are not simple brain teasers or abstruse word games—they challenge us to hone our reasoning and become more alert to the flaws in received wisdom and common habits of thought.

Lighthearted, witty, and conversational, Perplexing Paradoxes presents sophisticated material in an accessible way for all readers interested in the world’s boundless possibilities—and impossibilities.

There is a logical flaw in the statistical methods used across experimental science. This fault is not a minor academic quibble: it underlies a reproducibility crisis now threatening entire disciplines. In an increasingly statistics-reliant society, this same deeply rooted error shapes decisions in medicine, law, and public policy with profound consequences. The foundation of the problem is a misunderstanding of probability and its role in making inferences from observations.

Aubrey Clayton traces the history of how statistics went astray, beginning with the groundbreaking work of the seventeenth-century mathematician Jacob Bernoulli and winding through gambling, astronomy, and genetics. Clayton recounts the feuds among rival schools of statistics, exploring the surprisingly human problems that gave rise to the discipline and the all-too-human shortcomings that derailed it. He highlights how influential nineteenth- and twentieth-century figures developed a statistical methodology they claimed was purely objective in order to silence critics of their political agendas, including eugenics.

Clayton provides a clear account of the mathematics and logic of probability, conveying complex concepts accessibly for readers interested in the statistical methods that frame our understanding of the world. He contends that we need to take a Bayesian approach—that is, to incorporate prior knowledge when reasoning with incomplete information—in order to resolve the crisis. Ranging across math, philosophy, and culture, Bernoulli’s Fallacy explains why something has gone wrong with how we use data—and how to fix it.

There is a logical flaw in the statistical methods used across experimental science. This fault is not a minor academic quibble: it underlies a reproducibility crisis now threatening entire disciplines. In an increasingly statistics-reliant society, this same deeply rooted error shapes decisions in medicine, law, and public policy with profound consequences. The foundation of the problem is a misunderstanding of probability and its role in making inferences from observations.

Aubrey Clayton traces the history of how statistics went astray, beginning with the groundbreaking work of the seventeenth-century mathematician Jacob Bernoulli and winding through gambling, astronomy, and genetics. Clayton recounts the feuds among rival schools of statistics, exploring the surprisingly human problems that gave rise to the discipline and the all-too-human shortcomings that derailed it. He highlights how influential nineteenth- and twentieth-century figures developed a statistical methodology they claimed was purely objective in order to silence critics of their political agendas, including eugenics.

Clayton provides a clear account of the mathematics and logic of probability, conveying complex concepts accessibly for readers interested in the statistical methods that frame our understanding of the world. He contends that we need to take a Bayesian approach—that is, to incorporate prior knowledge when reasoning with incomplete information—in order to resolve the crisis. Ranging across math, philosophy, and culture, Bernoulli’s Fallacy explains why something has gone wrong with how we use data—and how to fix it.

Why does a clarinet play at lower pitches than a flute? What does it mean for sounds to be in or out of tune? How are emotions carried by music? Do other animals perceive sound like we do? How might a musician use math to come up with new ideas?

This book offers a lively exploration of the mathematics, physics, and neuroscience that underlie music in a way that readers without scientific background can follow. David Sulzer, also known in the musical world as Dave Soldier, explains why the perception of music encompasses the physics of sound, the functions of the ear and deep-brain auditory pathways, and the physiology of emotion. He delves into topics such as the math by which musical scales, rhythms, tuning, and harmonies are derived, from the days of Pythagoras to technological manipulation of sound waves. Sulzer ranges from styles from around the world to canonical composers to hip-hop, the history of experimental music, and animal sound by songbirds, cetaceans, bats, and insects. He makes accessible a vast range of material, helping readers discover the universal principles behind the music they find meaningful.

Written for musicians and music lovers with any level of science and math proficiency, including none, Music, Math, and Mind demystifies how music works while testifying to its beauty and wonder.

Why does a clarinet play at lower pitches than a flute? What does it mean for sounds to be in or out of tune? How are emotions carried by music? Do other animals perceive sound like we do? How might a musician use math to come up with new ideas?

This book offers a lively exploration of the mathematics, physics, and neuroscience that underlie music in a way that readers without scientific background can follow. David Sulzer, also known in the musical world as Dave Soldier, explains why the perception of music encompasses the physics of sound, the functions of the ear and deep-brain auditory pathways, and the physiology of emotion. He delves into topics such as the math by which musical scales, rhythms, tuning, and harmonies are derived, from the days of Pythagoras to technological manipulation of sound waves. Sulzer ranges from styles from around the world to canonical composers to hip-hop, the history of experimental music, and animal sound by songbirds, cetaceans, bats, and insects. He makes accessible a vast range of material, helping readers discover the universal principles behind the music they find meaningful.

Written for musicians and music lovers with any level of science and math proficiency, including none, Music, Math, and Mind demystifies how music works while testifying to its beauty and wonder.

Mathematics: The New Golden Age offers a glimpse of the extraordinary vistas and bizarre universes opened up by contemporary mathematicians: Hilbert's tenth problem and the four-color theorem, Gaussian integers, chaotic dynamics and the Mandelbrot set, infinite numbers, and strange number systems. Why a "new golden age"? According to Keith Devlin, we are currently witnessing an astronomical amount of mathematical research. Charting the most significant developments that have taken place in mathematics since 1960, Devlin expertly describes these advances for the interested layperson and adroitly summarizes their significance as he leads the reader into the heart of the most interesting mathematical perplexities—from the biggest known prime number to the Shimura-Taniyama conjecture for Fermat's Last Theorem.

Revised and updated to take into account dramatic developments of the 1980s and 1990s, Mathematics: The New Golden Age includes, in addition to Fermat's Last Theorem, major new sections on knots and topology, and the mathematics of the physical universe.

Devlin portrays mathematics not as a collection of procedures for solving problems, but as a unified part of human culture, as part of mankind's eternal quest to understand ourselves and the world in which we live. Though a genuine science, mathematics has strong artistic elements as well; this creativity is in evidence here as Devlin shows what mathematicians do—and reveals that it has little to do with numbers and arithmetic. This book brilliantly captures the fascinating new age of mathematics.

Despite the growing consensus on the need for action to counteract climate change, complex economic and political forces have so far prevented international actors from making much headway toward resolving the problem. Most approaches to climate change are based in economics and environmental science; in this book, Parkash Chander argues that we can make further progress on the climate change impasse by considering a third approach—game theory.

Chander shows that a game-theoretic approach, which offers insight into the nature of interactions between sovereign countries behaving strategically and the kinds of outcomes such interactions produce, can illuminate how best to achieve international agreements in support of climate-change mitigation strategies. Game Theory and Climate Change develops a conceptual framework with which to analyze climate change as a strategic or dynamic game, bringing together cooperative and noncooperative game theory and providing practical analyses of international negotiations. Chander offers economic and game-theoretic interpretations of both the Kyoto Protocol and the Paris Agreement and argues that the Paris Agreement may succeed where the Kyoto Protocol failed. Finally, Chander discusses the policy recommendations his framework generates, including a global agreement to support development of cleaner technologies on a global scale.