How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg

Jordan Ellenberg’s book “How Not to be Wrong: The Power of Mathematical Thinking” is a groundbreaking effort at popularizing mathematics and debunking the myth that those without formal mathematical training are doomed to be mathematicians. The book takes the reader through a history of mathematics, starting with Euclid’s ancient Greece and ending with modern theories, while also exploring real-world applications of mathematics. Rather than focusing on how to teach math, it focuses on how to apply math in everyday life.

The book begins by offering an alternative to popularized beliefs about mathematics and challenges assumptions about the nature of mathematical accuracy, such as the idea that being “absolutely certain” of something is an indicator of mathematical correctness. Throughout the book, Ellenberg compels the reader to think more deeply about their theories of accuracy by introducing concepts from mathematics. He then provides a systematic framework for understanding the power of mathematics and how it can be applied to real-world problems.

In addition, Ellenberg does not just provide theorems and proofs to mechanically apply; he provides models of thinking to broaden the reader’s understanding of the power of mathematics. For example, Ellenberg introduces the notion of "calibration" which involves assessing the accuracy of one’s own answer by testing it against others’. He applies this concept to examples to help the reader put this complex concept into practice.

The book also offers a detailed exploration of how mathematical thinking can affect our views of the world. Ellenberg examines how mathematics can help us understand how our world works and offers an alternative to the popular belief that mathematics is not relevant to the real world. He examines how mathematical thinking can be applied to education and politics, among other things.

In order to prove his point, Ellenberg spends a significant amount of time going over the concept of “causality.” He examines instances where mathematics is used to provide insight into causality such as in medical studies, economics and other fields. He then demonstrates how to apply mathematical thinking to answer questions involving causality.

The book ends by summarizing how mathematical thinking can be of help in dealing with life’s daily occurrences. Ellenberg provides tips on how to use mathematical thought processes and models of thinking to interactions with friends, family and colleagues. By applying mathematical thinking to these interactions, one can gain clarity and knowledge about the situations, which can lead to better decisions.

Overall, “How Not to Be Wrong” is an inspiring journey through the history of mathematics and a unique approach to the intersections between mathematics and the real world. Ellenberg offers readers the fundamentals of mathematical thinking and a better understanding of the power of mathematics. In doing so, Ellenberg demonstrates that mathematics have the potential to unlock sophisticated insights and to help solve real-world problems.