A Mathematician Reads the Newspaper

1. Mathematics Illuminates the News: A New Lens for Understanding

Mathematics is not primarily a matter of plugging numbers into formulas and performing rote computations. It is a way of thinking and questioning that may be unfamiliar to many of us, but is available to almost all of us.

Beyond Rote Calculation. The book champions mathematics as a powerful tool for critical thinking, not just a set of formulas. It encourages readers to question assumptions, analyze data, and understand the underlying structures that shape news stories. This approach moves beyond surface-level reporting to reveal deeper insights into social issues and popular culture.

Enhancing Critical Thinking. By applying mathematical concepts, readers can better evaluate the validity of claims, identify biases, and understand the limitations of predictions. This includes recognizing the difference between correlation and causation, understanding statistical significance, and appreciating the role of chance in various events. The book aims to empower readers to become more informed and discerning consumers of news.

Unveiling Hidden Structures. The author demonstrates how mathematical ideas can illuminate various aspects of news, from political power dynamics to economic forecasts. By providing a framework for analyzing quantitative, probabilistic, and dynamic facets of society, the book offers a revealing slant on the traditional "Who, What, Where, When, Why, and How" of journalism. This approach encourages readers to look beyond the surface and consider the underlying mathematical dimensions of the stories they encounter.

2. Voting and Power: Beyond Simple Majority Rule

There have been a number of schemes suggested to ensure that a group's power as measured by the Banzhaf index more closely reflects its percentage of the vote.

Challenging Conventional Voting. The book explores alternative voting systems beyond simple majority rule, highlighting the Banzhaf power index as a way to measure the actual influence of different groups or individuals. This index reveals how some voters can be dummies, whose votes never change the outcome, while others wield disproportionate power. The author uses the example of Lani Guinier's writings to illustrate the importance of understanding these power dynamics.

Cumulative Voting and Representation. Cumulative voting, where voters can distribute their votes among candidates, is presented as a way to empower minority groups and ensure fairer representation. This system allows cohesive groups to concentrate their votes on a single candidate, increasing their chances of electing someone who represents their interests. The author suggests that such tinkering with election procedures is not unheard of and can lead to more equitable outcomes.

Voting Anomalies and Paradoxes. The book acknowledges that every voting method has undesirable consequences and fault lines, referencing Arrow's theorem. It emphasizes that the question is not whether we should be democratic, but how we should be democratic, advocating for an open experimental approach to electoral systems. The author encourages readers to consider the mathematical roots of voting systems and recognize that there are many different styles of democracy, all with their own imperfections.

3. Psychological Biases: Availability, Anchoring, and the Media

Models that are fitting for one domain are frequently inappropriate for another that is superficially similar.

The Availability Heuristic. The book delves into the availability error, a psychological phenomenon where judgments are heavily influenced by the first thing that comes to mind. This bias can lead to misinterpretations of news stories, as readers tend to make evaluations based on easily recalled or emotionally evocative information. The author provides examples such as the overestimation of words starting with "r" and the impact of memorized words on evaluating ambiguous news stories.

Anchoring Effects and Framing. Anchoring effects, where initial figures or information heavily influence subsequent estimates, are also discussed. The book illustrates how the wording of poll questions can significantly alter responses, and how the framing of a story can shape public perception. The author emphasizes the importance of consciously searching for alternative interpretations to counter these biases.

Combating Misinformation. To avoid being taken in by facile parallels and analogies, the book suggests readers consciously search for interpretations that undermine the prevailing one. It encourages critical thinking by asking how events are actually different, how stories can be put into a more neutral context, and what is crucial versus accidental about them. The author stresses that while searching for resemblances is natural, it's important to avoid simplistic analogies and be aware of the halo effect, where judgments are based on one salient characteristic.

4. Economic Forecasting: Embracing Unpredictability and Chaos

You can only predict things after they've happened.

The Illusion of Predictability. The book critiques the common assumption that economic and political matters are predictable, highlighting the fatuous nature of much commentary and forecasting. It uses Arthur Laffer's Laffer curve as an example of oversimplified economic analysis, pointing out that the interconnectedness of variables is often ignored. The author argues that many social forecasts are either hollow or disguised with sophisticated language.

Nonlinear Dynamics and Chaos Theory. The concept of nonlinear dynamical systems is introduced to model the interconnections of economic variables. The book uses the analogy of a pool table with obstacles to illustrate how minuscule variations can lead to vastly different outcomes. The Butterfly Effect, where small changes in initial conditions can have disproportionate effects, is also discussed.

Skepticism and Awareness. The book advises readers to be wary of glib and simplistic diagnoses, emphasizing the imprecision and unreliability of standard economic statistics. It suggests observing stories where small differences seem magnified, articles about self-fulfilling prophecies, and accounts pointing to single causes or immediate consequences. The author encourages readers to take seriously only those predictions that are short-term, deal with simple phenomena, and are not colored by the participants' intentions.

5. News Structures: Inverted Pyramids and Headline Realities

The advantage of this structure is that the story may be truncated anywhere, its end cut off for typographic or other reasons, and what remains still makes sense.

The Inverted Pyramid Structure. The book examines the inverted pyramid structure of news stories, where the most important information is presented first, followed by details in descending order of importance. While this structure allows for easy truncation and quick consumption of information, it also tends to obscure complexity and rely heavily on the reader's background assumptions. The author notes that this structure can lead to stories pandering to popular misunderstandings or stressing conflict between ethnic groups.

Attention Spans and Encapsulation. The author argues that newspaper stories, with their catchy headlines and information-filled leads, are more likely to create short attention spans than television. The ability to skim a newspaper and get the gist of dozens of stories contrasts with the long but shallow attention span required by television. The book suggests that encapsulation, while necessary for brevity, can oversimplify complex issues.

Alternative Storytelling Approaches. The book proposes a compromise shape that combines the need for quick communication of essential facts with a more leisurely fleshing out of these facts: a small inverted pyramid followed by a longer diamond-shaped story with a proper buildup and denouement. The author encourages readers to analyze headlines and compare them with the actual content of the stories, highlighting the potential for discrepancies and misrepresentations.

6. Probability and Politics: Games, Gambles, and Rationality

It's an intriguing consequence of the mathematical discipline of game theory that the conscious randomizing of choices can, if done right, maximize one's effectiveness.

Game Theory and Political Strategy. The book explores the application of game theory to political decision-making, suggesting that conscious randomization of choices can maximize effectiveness. It uses the example of a baseball pitcher choosing between fast balls and curve balls to illustrate the need for probabilistic strategies. The author argues that political leaders facing trade talks or other negotiations must sometimes let their positions be dictated by a roll of the dice to maximize their advantage.

Acknowledging Uncertainty. The book emphasizes the value of acknowledging uncertainty and behaving accordingly, using the example of a dial whose pointer stops on red 70% of the time and green 30% of the time. It points out that most people incorrectly vary their predictions, while the optimal strategy is to always predict red. The author suggests that politicians who abjure unwarranted expressions of certitude deserve plaudits, not pillory.

Rationality and Irrationality. The author finds it oddly liberating that rational and irrational decision-making processes are sometimes indistinguishable. The book encourages readers to recognize the role of chance in various situations and to adopt unvarying policies when appropriate. It advocates for politicians who acknowledge uncertainty and avoid unwarranted expressions of certitude.

7. Statistical Disparities: Beyond Simple Accusations of Bias

Such statistical disparities are not necessarily evidence of racism or ethnic prejudice, although, without a doubt, they sometimes are.

Normal Distributions and Extremes. The book discusses how even slight differences in the averages of two population groups can be accentuated at the extreme ends of normal bell-shaped curves. This means that even if the average height of one group is only slightly greater than the average height of another, people from the taller group will constitute a large majority among the very tall. The author cautions against drawing immediate conclusions of bias from such statistical disparities.

The Impossibility of Strict Proportionality. The book argues that schemes of strict proportional representation are impossible to implement, using the example of a company trying to create a workforce that "fairly" reflects the community. It highlights the complexities of intersecting group memberships and the unlikelihood of uniform backgrounds and training across these various cross-sections. The author emphasizes that statistical disparities will necessarily result.

Focus on the Individual. The book asserts that the basic unit upon which any liberal society is founded is the individual, not the group. It argues that racism, homophobia, and all other forms of group hatreds are real enough without making them our unthinking first inference when confronted with statistical disparities. The author encourages readers to debate whether tests are appropriate for their purpose, but not to be surprised when normal curves behave normally.

8. The Illusion of Control: Chance, Intention, and the News

The job of a newspaper reporter is to tell stories and to emphasize the role of the agent in any narrative.

The Role of Chance. The book challenges the journalistic emphasis on the role of the agent in narratives, arguing that the real story is often as dependent on chance as on intention. It questions what we would think of a political leader who made important decisions by a flip of the coin, suggesting that such a leader might be more rational than we assume. The author introduces the mathematical discipline of game theory to illustrate how conscious randomizing of choices can maximize one's effectiveness.

The Spinner Experiment. The book describes a probabilistic situation where people are asked to predict the outcome of a dial spin that stops on red 70% of the time and green 30% of the time. It points out that most people act as if they have some insight into the spinner's patterns, while the optimal strategy is to always predict red. The author emphasizes the value of acknowledging uncertainty and behaving accordingly.

Acknowledging Uncertainty. The book suggests that politicians who abjure unwarranted expressions of certitude deserve plaudits, not pillory. It encourages readers to acknowledge the role of chance in various situations and to adopt unvarying policies when appropriate. The author stresses the importance of distinguishing between skill and luck, and of recognizing the limitations of our ability to control outcomes.

9. Self-Reference and the Media: A Tangled Web of Influence

L i k e news content, the frequency of reporting on various newsmakers also has an inverted-pyramid structure.

The Inverted Pyramid of Newsmakers. The book observes that the frequency of reporting on various newsmakers also follows an inverted-pyramid structure, with important newsmakers making vastly more news than unimportant ones. It identifies the president of the United States as the number-one newsmaker, followed by presidential candidates, congressional representatives, and other federal officials. The author notes that reporters naturally gravitate to where news is made, which is often Washington, D.C.

Zipf's Law and Newsmakers. The book draws a parallel between the inverted-pyramid structure of newsmakers and Zipf's Law in linguistics, which states that the frequency of a word is inversely proportional to its rank order. It conjectures that there may be a similar relationship between the frequency of occurrence of a particular newsmaker and his or her rank order of occurrence. The author suggests that the president would be the analogue of the word "the" in American newspapers.

The Power of Language and Style. The book includes a little experiment performed by the author that is germane to the issue of newspapers and linguistics. After determining the grammatical characteristics of a few tabloid news stories, the author wrote a story that syntactically conformed to them. The author suggests that with more powerful and intelligent software, one could easily approach the sound (but not the sense) of more substantive pieces.

10. The Complexity Horizon: Limits to Understanding and Prediction

The sensitive dependence of nonlinear systems on their initial conditions has been called the Butterfly Effect, from the idea that a butterfly flapping its wings in China, say, might spell the difference several months later between a hurricane and a balmy day along the eastern U.S. seaboard.

Nonlinear Systems and the Butterfly Effect. The book introduces the concept of nonlinear dynamical systems and the Butterfly Effect, where small changes in initial conditions can lead to vastly different outcomes. It uses the example of a billiard table to illustrate how minuscule variations in the angle of impact can be magnified by successive hits of the obstacles. The author emphasizes that the economy and other social systems are similarly sensitive to initial conditions.

Fractals and Complexity. The book discusses fractals as astoundingly intricate and self-similar curves (or surfaces, solids, or higher-dimensional objects) that contain more, but similar, complexity the closer one looks. It cites examples such as shorelines, mountains, and clouds to illustrate the fractal nature of many shapes in nature. The author suggests that these nonlinear systems demonstrate a complex unpredictability that seems to arise even when the systems are defined by quite elementary, nonlinear rules and equations.

Wariness of Simplistic Diagnoses. The book advises readers to be wary of glib and simplistic diagnoses, given the interconnectedness of variables and the Butterfly Effect. It suggests that the accuracy of social forecasts and predictions is vastly greater if the predictions are short-term rather than long-term, if they deal with simple rather than complex phenomena, and if they are not colored by the participants' intentions. The author emphasizes that skepticism should guide us when we read about any political, economic, or military policy of any complexity.

11. Science Reporting: Balancing Accuracy, Advocacy, and Skepticism

Models that are fitting for one domain are frequently inappropriate for another that is superficially similar.

The Need for Context. The book argues that news stories are often bereft of the numbers that would enable a reader to put them in perspective, particularly in times of war. It cites the example of Iraqi military casualties during the Gulf War, noting that little effort was expended to come up with casualty figures and that these figures were rarely reformulated and put in perspective. The author emphasizes the importance of benchmark figures in multicultural societies where a common culture, stories, and myths are not as widely shared.

Challenging Absurd Claims. The book encourages reporters to challenge absurd claims, such as the assertion that 600 million African-American deaths were attributable to slavery or that 75% of American slaves were owned by Jews. It suggests that similar elementary research could demolish such claims. The author emphasizes the importance of including round figures in stories and of comparing numbers with quantities that are more viscerally appreciated.

The Dangers of Oversimplification. The book cautions against the tendency to be taken in by facile parallels and analogies, emphasizing the importance of complex coverage. It suggests that a weaker but more practical antidote is consciously to search for interpretations or associations that undermine the prevailing one that is so temptingly available. The author stresses the need for journalists to avoid bolstering the conventional wisdom and to be aware of the halo effect and anchoring effects.

12. The Power of Numbers: From Health Risks to Environmental Concerns

You can only predict things after they've happened.

The Art of Numerical Craftiness. The book explores how descriptions of numerical relations depend on their authors' intentions, noting that consumer groups, political groups, and business advertisers might stress a linear measure of size to make a quantity appear large, or stress its volume to make it appear small. It highlights the importance of knowing when to redefine terms, when to interpret them loosely, and when to take them literally. The author uses the example of President Clinton's defense of his wife's integrity to illustrate the artful interpretation that is the most characteristic aspect of political discourse.

Benchmark Figures and Context. The book emphasizes the importance of including appropriate benchmark estimates in articles on economics and other topics. It suggests that mentioning the fact that the annual GNP is approximately $6 trillion serves to orient the reader. The author also advocates for comparing numbers with quantities that are more viscerally appreciated, such as the cost of the savings-and-loan bailout.

The Importance of Numeracy. The book argues that readers and reporters should have the rudimentary numeracy to be able to translate from rates to absolute numbers and back, or to convert from one unit to another. It stresses the need for basic factual knowledge and appropriate definitions to make reasonable estimates. The author concludes that unless we find better ways to vivify complex issues and to keep our heads in times of crisis, the cost to us will continue to be much more than $500 billion.

Last updated: February 21, 2025