Books like Gamma — picked by fans

David Hilbert was the most important mathematician at the dawn of the 20th century. In 1900, he gave the mathematical community its homework for the next 100 years setting out the list of open problems that had to be solved by 2000. While to the rest of the mathematicians, he may have appeared as their professor, he was also the class clown. As colorful and funny as he was brilliant, you cannot but come away loving this great mathematical genius.

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.

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.

Mathematicians are constantly baffled by the public’s lack of awareness, not just of what mathematics does, but what it is. Today’s technological society functions only because of a vast range of mathematical concepts, techniques, and discoveries, which go far beyond elementary arithmetic and algebra. This was one of the first books to tackle these misunderstandings head on. It does so by examining not just the math and what it’s used for, but the social structures, the ‘conditions of civilization’ that have brought us to this curious state: utterly dependent on math, almost universally unaware that we are. 

What child isn’t curious about the night sky and all the stars that live up there? Did you know that the Sun is a giant star?  This book is full of fun facts, not just about stars but about our planet. It helps to put things into perspective, so to speak. It talks about gravity and how many miles away the moon is from the earth. I think kids will learn a lot from reading this book and will even be able to impress their friends with all of their newly acquired knowledge. Did you know the earth looks green because it’s covered in 3,000,000,000,000 trees?? I love this book because learning new things is fun and this book is all about fun. I felt very smart after reading this book.

I love this book because it illustrates how a deep understanding of one subject area (mathematics) can help develop a deeper understanding of another (biology). This is a common theme in my teaching. I love how the author links the two fields of mathematics and biology in a way that is accessible to people in both fields.

This book will extend the knowledge of anyone working in the interdisciplinary area of mathematical biology and encourage the reader to see the value of viewing the world through interdisciplinary eyes. 

Many undergraduate mathematics books – even those aimed at new students – are dense, technical, and difficult to read at any sort of speed. This is a natural feature of books in a deductive science, but it can be very discouraging, even for dedicated students. Houston’s book covers many ideas useful at the transition to proof-based mathematics, and he has worked extensively and attentively with students at that stage. Consequently, his book maintains high mathematical integrity and has lots of useful exercises while also being an unusually friendly read.

Mathematics requires accurate calculation, and students sometimes think that getting the right answer is enough. But mathematics is also about valid logical arguments, and the demand for clear communication increases through an undergraduate degree. Students, therefore, need to learn to write professionally, with attention to general issues like good grammar, and mathematics-specific issues like accuracy in notation, precision in logical language, and structure in extended arguments. Vivaldi’s book has a great many examples and exercises, and students could benefit from studying it systematically or from dipping into it occasionally and reflecting on small ways to improve.

The Univalent Foundations program in foundations of mathematics launched by Voevodsky and others in the past decade and a half has contributed to a promising new paradigm unifying computation, mathematics, logic, and proof theory.

Understanding the core elements of this research program, Homotopy Type Theory, is essential for contemporary philosophers who want to engage directly with current developments in mathematics and computer science.

Corfield is a well-established name in philosophy of mathematics, and this book is the best introduction to Homotopy Type Theory for philosophers.

Working within themes and problematics that will be familiar to philosophers with a basic background in logic, Corfield covers the elementary constructions of homotopy types from a logical point of view and provides plenty of provocative suggestions for how these formal tools might reinvigorate philosophical research today.