The best books for mathematical inspiration

Thomas A. Garrity Author Of All the Math You Missed: (But Need to Know for Graduate School)
By Thomas A. Garrity

Who am I?

I love mathematics and truly believe that “Functions Describe the World.” I'm deeply satisfied that I've spent my professional life discovering new mathematics and explaining known mathematics to others. I was an undergraduate at the University of Texas, Austin, got my PhD from Brown University, then spent three years as a G.C. Evans Instructor at Rice University, before moving to Williams, where I've been ever since. Besides writing All the Math You Missed (But Need to Know for Graduate School), I've also written Algebraic Geometry: A Problem Solving Approach (with a number of co-authors) and Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell’s Equations to Yang-Mills, and a number of research articles.  

I wrote...

All the Math You Missed: (But Need to Know for Graduate School)

By Thomas A. Garrity,

Book cover of All the Math You Missed: (But Need to Know for Graduate School)

What is my book about?

People who are starting graduate school in mathematics are full of hopes and dreams to become great mathematicians. That is good. But most are suddenly confronted with the cold hard fact that they are expected to know a daunting breadth of mathematics, a breadth that few actually have or even could have had. This book is an attempt to help my younger future colleagues. 

Each of its twenty chapters covers a key part of the math needed for graduate school. All beginning graduate students know the math in some of the chapters. Hardly any are comfortable with the material in all of the chapters. This book will help them “get into the game,” concentrating on why the math in each chapter is important and pointing them to resources to learn more. 

The Books I Picked & Why

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By Constance Reid,

Book cover of Hilbert

Why this book?

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.  

Adventures of a Mathematician

By S. M. Ulam,

Book cover of Adventures of a Mathematician

Why this book?

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. 

Prime Obsession: Berhhard Riemann and the Greatest Unsolved Problem in Mathematics

By John Derbyshire,

Book cover of Prime Obsession: Berhhard Riemann and the Greatest Unsolved Problem in Mathematics

Why this book?

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: Exploring Euler's Constant

By Julian Havil,

Book cover of Gamma: Exploring Euler's Constant

Why this book?

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. 

Euclid's Elements

By Dana Densmore (editor), Thomas L. Heath (translator),

Book cover of Euclid's Elements

Why this book?

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.

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