Introduction
In "How Not to Be Wrong," mathematician Jordan Ellenberg takes readers on an enlightening journey through the world of mathematics and its profound impact on our daily lives. Far from being a dry, academic subject confined to classrooms, Ellenberg argues that mathematics is the science of common sense, a reflection of our intuitive understanding of the world around us.
This book aims to uncover the "hidden math" in our lives and demonstrate how mathematical thinking can help us make better decisions, avoid common pitfalls, and ultimately, learn not to be wrong. Ellenberg's approach is refreshing and accessible, breaking down complex concepts into digestible pieces that even those who struggled with math in school can appreciate.
The Essence of Mathematics
Mathematics as the Science of Not Being Wrong
At its core, mathematics is about using reason to avoid being wrong about things. It's a powerful tool for solving common problems, even when we don't explicitly recognize that we're using math. Ellenberg illustrates this point with a fascinating example from World War II.
During the war, American planes returning from missions in Europe were often riddled with bullet holes. Military advisors, noticing that the fuselage had more bullet holes than the engine, suggested reinforcing the fuselage with better armor. However, a young mathematician proposed a counterintuitive solution: reinforce the engine instead.
This mathematician recognized a phenomenon known as survivorship bias. He realized that the planes that took shots to the engine were likely the ones that didn't make it back. By focusing on the surviving planes, the advisors were overlooking a crucial piece of information. This example demonstrates how mathematical thinking can lead to more accurate conclusions and better decision-making.
Mathematics and Common Sense
Ellenberg argues that mathematics is fundamentally based on common sense. Many mathematical principles reflect intuitive truths that we understand without formal training. For instance, the commutative property of addition (a + b = b + a) is something we instinctively know, even if we can't articulate it mathematically.
This connection between math and intuition is crucial because it shows that mathematical thinking isn't as foreign or inaccessible as many people believe. Instead, it's an extension and formalization of our natural reasoning processes.
The Power of Linearity
Simplifying Complex Problems
One of the key strategies in mathematics is to simplify difficult problems by breaking them down into more manageable parts. This often involves assuming linearity, which allows us to work with straight lines and simple relationships instead of complex curves.
Ellenberg uses the example of an ant walking around a circle to illustrate this concept. From the ant's perspective, the circle's curve appears to be a straight line. Similarly, if we zoom in closely enough on any curve, it begins to look like a straight line. This principle allows mathematicians to approximate complex shapes and relationships using simpler, linear components.
Linear Regression in Statistics
The concept of linearity is widely applied in statistics through a technique called linear regression. This method is used to analyze relationships between variables, such as the correlation between income levels and voting preferences. Linear regression simplifies these relationships by representing them as straight lines, providing a useful approximation of trends in data.
While this simplification can be incredibly useful, Ellenberg cautions that it's important to recognize when linearity assumptions are appropriate and when they might lead to misleading conclusions.
The Challenges of Observational Data
The Pitfalls of Chance
Drawing conclusions from observational data can be tricky, as patterns and correlations can sometimes arise by chance. Ellenberg humorously illustrates this point with the story of a neuroscientist who showed photos to a dead fish and measured its brain activity. Surprisingly, the fish appeared to respond accurately to the emotions in the pictures.
This absurd experiment highlights how easily research findings can be influenced by random noise, especially when dealing with large amounts of data. In the case of brain scans, the sheer number of data points (voxels) increases the likelihood that some will show a correlation with the stimulus purely by chance.
Probability Theory and Significance Testing
To address the issue of chance correlations, scientists use probability theory and statistical significance testing. The null hypothesis significance test is a common tool used to determine whether observed data is likely to have occurred by chance.
Ellenberg explains this concept using the example of testing a new drug. Scientists start with a null hypothesis (in this case, that the drug has no effect) and then calculate the probability that their observed data could have occurred if the null hypothesis were true. If this probability (p-value) is below a certain threshold (usually 0.05), the results are considered statistically significant.
While this approach helps filter out many chance occurrences, Ellenberg notes that it's not foolproof and can still lead to false conclusions if not used carefully.
Understanding Probability and Risk
Expected Value in Betting
Probability theory is a powerful tool for understanding uncertain situations, such as betting or gambling. Ellenberg introduces the concept of expected value, which helps us determine what we should expect from a bet on average.
Using the example of a lottery ticket, Ellenberg demonstrates how to calculate expected value by considering all possible outcomes, their probabilities, and their associated values. This calculation reveals that most lotteries have a negative expected value, meaning that on average, players lose money.
The Importance of Risk Assessment
While expected value is useful, Ellenberg emphasizes that it doesn't tell the whole story. Risk assessment is equally important, especially when dealing with high-stakes situations. He illustrates this with a thought experiment: Would you rather receive $50,000 or take a 50/50 bet between losing $100,000 and gaining $200,000?
Although both options have the same expected value, the potential consequences of losing the bet are much more severe. This example highlights the importance of considering not just the average outcome, but also the potential risks and one's ability to absorb losses.
The Regression Effect
Understanding Regression to the Mean
Ellenberg introduces the concept of regression to the mean, a statistical phenomenon that occurs when extreme outcomes are followed by more average ones. This effect can be observed in various fields, from sports performance to literary success.
For example, the children of very tall parents tend to be tall, but not as tall as their parents. Similarly, a novelist's second book is often perceived as less impressive than their breakout debut. These occurrences aren't due to some mystical force, but rather a natural statistical tendency for extreme results to be followed by more average ones.
Misinterpreting the Regression Effect
The regression effect is often misunderstood or overlooked, leading to false conclusions. Ellenberg cites a study on bran's effects on digestion as an example. The researchers attributed changes in digestion rates to bran consumption, when in reality, the observed effect was likely just regression to the mean.
This misinterpretation of the regression effect is common in various fields, from medicine to sports coaching. Recognizing this phenomenon can help us avoid drawing incorrect conclusions from data and making misguided decisions based on those conclusions.
The Limitations of Linear Regression
When Linearity Assumptions Fail
While linear regression is a powerful statistical tool, Ellenberg cautions against its misuse. Linear regression is only appropriate when the relationship between variables is genuinely linear. Applying it to non-linear relationships can lead to absurd predictions and false conclusions.
Ellenberg illustrates this with the example of a missile's trajectory. While linear regression might accurately predict the missile's position in the short term, it fails to account for the curved path over longer distances. Similarly, he critiques a study that used linear regression to predict that all Americans would be obese by 2048, pointing out that such trends tend to follow curved rather than straight lines over time.
The Importance of Appropriate Model Selection
This discussion underscores the importance of choosing appropriate mathematical models for the phenomena we're studying. While simplification is often necessary and useful, it's crucial to recognize when our simplifying assumptions (like linearity) break down. Ellenberg encourages readers to think critically about the models used in studies and predictions they encounter, considering whether the underlying assumptions are reasonable.
The Prevalence of False Research Findings
Statistical Significance and False Positives
Ellenberg delves into the controversial topic of why many published research findings may be false. He cites the work of John Ioannidis, who argued that the way statistical significance is typically used in research can lead to a high rate of false positives.
One key issue is that when researchers test a large number of hypotheses, some will appear statistically significant purely by chance. For example, in genetic studies looking for links to a particular disease, thousands of genes might be tested. Even if only a small percentage show a significant correlation by chance, this could still result in hundreds of false positives.
Publication Bias and Data Manipulation
Another problem Ellenberg highlights is publication bias. Studies that find significant results are more likely to be published than those that don't, leading to a skewed representation of research in scientific literature. This can make certain effects appear more robust or widespread than they actually are.
Ellenberg also discusses the temptation for researchers to manipulate their data or analysis methods to achieve statistical significance. While this isn't necessarily done with malicious intent, it can lead to the publication of findings that don't hold up under scrutiny.
These issues underscore the importance of critical thinking when interpreting scientific studies and highlight the need for more robust statistical practices in research.
The Complexities of Public Opinion
The Challenges of Measuring Public Opinion
In the final section of the book, Ellenberg turns his attention to the difficulties of accurately measuring and representing public opinion. He argues that polls and elections often oversimplify complex issues and can lead to misleading conclusions about what "the public" thinks.
One key problem is that people's opinions are often contradictory or context-dependent. Ellenberg cites polls about government spending where respondents simultaneously supported cutting overall spending while increasing spending in most specific categories. These contradictions highlight the complexity of public opinion and the limitations of simple yes/no questions in capturing it.
The Pitfalls of Majority Rule
Ellenberg also critiques the concept of "majority rule" in elections and polls. He points out that when there are more than two options, the way choices are grouped can dramatically affect the outcome. For example, breaking down opposition to a policy into different categories (e.g., "repeal," "weaken," "leave as is," "expand") can reveal a more nuanced picture than a simple for/against split.
The 2000 U.S. presidential election serves as a powerful example of how voting systems can fail to accurately represent public preferences. While George Bush won Florida by a narrow margin over Al Gore, the presence of a third candidate, Ralph Nader, likely affected the outcome in a way that didn't reflect the majority's preference.
These examples underscore the importance of looking beyond simple majorities and headlines when interpreting polls and election results. They also highlight the need for more sophisticated voting systems that can better capture the complexity of public opinion.
Conclusion: The Value of Mathematical Thinking
Throughout "How Not to Be Wrong," Jordan Ellenberg makes a compelling case for the importance of mathematical thinking in everyday life. By uncovering the hidden math in various situations – from wartime decision-making to interpreting scientific studies and understanding public opinion – he demonstrates how mathematical reasoning can lead to better decisions and a clearer understanding of the world around us.
Key takeaways from the book include:
Mathematics is not just about numbers and formulas, but about logical reasoning and avoiding errors in thinking.
Many mathematical concepts are based on common sense and intuition, making them more accessible than they might first appear.
Simplifying complex problems through techniques like linear regression can be powerful, but it's crucial to recognize the limitations of these simplifications.
Understanding probability and risk is essential for making informed decisions in uncertain situations.
Statistical phenomena like regression to the mean and survivorship bias can lead to misinterpretations if not properly understood.
Critical thinking is necessary when interpreting scientific studies, polls, and other data-driven claims.
Complex issues like public opinion often resist simple quantification and require nuanced analysis.
Ellenberg's book serves as both an introduction to important mathematical concepts and a guide to applying these concepts in real-world situations. By developing our mathematical thinking skills, we can become better equipped to navigate the complexities of modern life, make more informed decisions, and ultimately, learn how not to be wrong.
The author's engaging style and use of relatable examples make the book accessible to readers regardless of their mathematical background. "How Not to Be Wrong" challenges us to think more critically about the numbers and statistics we encounter daily, encouraging a deeper engagement with the mathematical underpinnings of our world.
In an era where data and statistics play an increasingly prominent role in public discourse and decision-making, the skills and insights offered by Ellenberg's book are more valuable than ever. By embracing mathematical thinking, we can become more discerning consumers of information, more effective problem-solvers, and better equipped to understand and shape the world around us.
Ultimately, "How Not to Be Wrong" is not just about avoiding errors; it's about developing a more nuanced, analytical approach to life's challenges and opportunities. It reminds us that mathematics is not a distant, abstract discipline, but a powerful tool for understanding and improving our everyday experiences. By learning to think like a mathematician, we can enhance our ability to reason, make decisions, and navigate the complexities of our increasingly data-driven world.