Book review: How Not to Be Wrong. The Power of Mathematical Thinking

Why this Book?

Mathematics is something that either brings back bad memories or something you might enjoyed early on but failed to see it’s relevance in real life. “How Not to Be Wrong” chooses to challenge this perception, revealing mathematics as a vibrant, practical tool for understanding the world around us.

Jordan Ellenberg explains problems where math isn’t confined to classrooms or academic papers, it’s woven into the fabric of everyday life, from the lottery tickets we buy to the medical studies we read about, from political polls to questions about the existence of God.

The Power of Mathematical Thinking

Ellenberg opens with a striking World War II story about Abraham Wald, a mathematician who solved a critical problem for the Air Force. Planes returning from missions showed bullet hole patterns concentrated in the fuselage. Officers wanted to add armor where the bullet holes were. Wald’s insight was unique, arguing Armor the places where the bullet holes weren’t, because planes hit in those areas never made it back (Survivorship Bias).

This is the essence of mathematical thinking: seeing what’s not there, understanding hidden patterns, and questioning our first instincts.

The author uses this foundation to explore how mathematical reasoning helps us avoid common pitfalls. Whether it’s understanding why tall parents tend to have shorter children (regression to the mean), or why a successful pilot study often disappoints when scaled up, Ellenberg shows how math provides a framework for not being wrong.

Making the Invisible Visible

One of the book’s strengths is how it makes abstract concepts tangible. Ellenberg doesn’t shy away from the actual mathematics, showing readers the equations and logic while explaining them in clear, accessible language.

He takes on topics like statistical significance, showing how the 95% confidence threshold that scientists use can be easily manipulated to produce misleading results. Through examples like testing multiple hypotheses, he demonstrates how “statistically significant” results can emerge purely by chance.

The book delves into probability, correlation versus causation, expected value, and Bayesian inference, all through engaging stories and real-world examples that stick with you long after you’ve finished reading.

The Lottery as Mathematical Theater

Perhaps the most entertaining section involves groups of mathematically savvy individuals who actually made money from state lotteries. Not through luck, but through careful analysis of the rules and odds.

Ellenberg explains how certain lottery games were designed with flaws that created situations where, if you bought enough tickets at the right time, you had a positive expected value. These weren’t scams or illegal schemes, they were simply people understanding the mathematics better than the lottery designers did.

It’s a fascinating exploration of how mathematical thinking can turn a game of chance into a predictable investment. And it includes the delightful detail that Voltaire himself made his fortune through similar lottery arbitrage in 18th-century France.

Numbers Don’t Lie, But They Can Mislead

Medical studies can produce statistically significant results that are nevertheless meaningless. Polls can show clear majorities while missing the complexity of public opinion.

Ellenberg dedicates attention to election mathematics, dissecting how different mathematical approaches to the same data can yield different conclusions. He’s interested in showing how certainty is often elusive even when we have numbers on our side.

The book also tackles the replication crisis in psychology, showing how the combination of publication bias and p-value hacking has led to a literature full of findings that don’t hold up under scrutiny.

Understanding Uncertainty

One of the most valuable lessons for me is about embracing uncertainty rather than demanding false certainty. Ellenberg discusses probabilistic forecasting, showing how explicitly including margins of error and confidence intervals leads to better predictions than false precision.

The book argues persuasively that good mathematical thinking means acknowledging what we don’t know, quantifying our uncertainty, and being willing to update our beliefs when new evidence arrives. This approach to knowledge stands in stark contrast to the dogmatic certainty that pervades much of public discourse.

Conclusion

“How Not to Be Wrong” is a book about thinking clearly in a world that often rewards confident nonsense over careful reasoning. It shows that mathematics isn’t about memorizing formulas—it’s about developing mental tools for seeing through the surface chaos to the underlying structure.

Ellenberg writes with obvious passion for his subject, but never comes across as condescending. He knows that mathematics can be hard, that our intuitions often fail us, and that even professional mathematicians get things wrong. The goal isn’t perfection; it’s using mathematical reasoning to be wrong less often.

For anyone who wants to think more critically about statistics they see in the news, make better decisions under uncertainty, or simply appreciate the hidden mathematical beauty in everyday phenomena, this book is an excellent guide. It’s challenging at times, you’ll need to engage with the ideas, not just skim them, but the reward is a richer understanding of how the world works and how we can navigate it more wisely.