The best books for mathematical inspiration

David Hilbert was one of the great mathematicians of the early twentieth century. He also created an entire research environment at the University of Göttingen, founded on the fundamental assumption that there is a deep unity behind all of mathematics (an assumption that in part motivated me to write All the Math You Missed). From this school much of the mathematical triumphs of the last 100 years have sprung (especially from the revolutionary work rotating around the mathematics of Emmy Noether in the 1920s in Göttingen). At least that is my impression from reading this book. It inspires young mathematicians to believe that it is indeed possible that “mathematics is the ultimate description of reality.” It certainly had that effect on me as a college junior worrying about my future life.  

Ulam was a Polish mathematical prodigy, publishing significant mathematics by the time he was 20. He was part of the rich Polish math community centered around Stefan Banach. Unlike most, he was heading to the United States in 1939 (with his younger brother) when Germany invaded Poland. All the rest of his family were murdered by the Nazis. He on the other hand ended up in Los Alamos, providing critical help on the Manhattan Project. Later in life, he wrote this book, his autobiography. Based on his history, one could well think that it would be a book full of tragic grief. Instead, it is a pean to the joys of doing mathematics and of living a life full of mathematics, without downplaying the horrors of the mid-twentieth century. 

Most mathematicians believe that the Riemann Hypothesis is the most important open question in mathematics, including me. But it is almost impossible to explain why this is such a central concern. This book is one of the attempts to explain to the non-mathematician why the Riemann Hypothesis is so important. As a partial spoiler alert, it has to do with the nature of prime numbers, which in part explains the title. It is not a book to read in one sitting, but it with a little work is great for seeing, at least in part, the big picture.

Gamma is a number, though little understood. Even its most basic properties are still unknown. We don’t even know if it is a rational number (a ratio of integers). This wonderful book explains why anyone would care. While it does require some mathematical background, anyone who has had calculus and is willing to read the book with a notepad and pen next to them in order to check and explore the formulas on their own will come away with a true appreciation of gamma and its impact. 

This is the bestselling textbook of all time. Euclid’s Elements has been the model for correct thinking for thousands of years. The traditional year-long course on axiomatic reasoning about geometry was easily my favorite course in high school. In fact, I sort of assumed that I was slow in that I could not “see” the underlying axioms in other classes. I simply did not realize that the other high school subjects were not axiom-based. 

The story goes that as a young prairie lawyer, Abraham Lincoln carried around with him a tattered copy of the Elements so that he could learn how to think (even though he never really had much formal education). I hope this is true. Even more so, I want to believe that he developed his profound oratorial skill from the power behind Euclid.