100 books like How to Think Like a Mathematician

I think about this book every day, even though it was written almost 25 years ago, and the edition I read explained how to manage your paper file folders! (One of my most-used apps, the to-do manager Things, is built on this system.)

I love how much time this book has saved me as I juggle running several businesses, staying active in my hobbies, and running a household. Allen’s approach to capturing your ideas and then deciding how to organize them so that you can keep track of what needs your attention is both simple and really profound.

For athletes who need to be as efficient as possible to reserve time and energy for training, this book is a lifesaver.

This book provides a systematic account of how to understand and structure mathematical proofs. Its approach is almost entirely syntactic, which is the opposite of how I naturally think – I tend to generate arguments based on examples, diagrams, and conceptual understanding. But that difference, for me, is precisely what makes this book so valuable. Solow gives a no-nonsense, practical, almost algorithmic approach to interpreting logical language and to tackling the associated reasoning. His book thereby provides the best answer I know of to the “How do I start?” problem so often encountered when students begin constructing proofs.  

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.

Research in cognitive psychology has revealed a lot about human learning and how to make it more effective. Most mathematics students – and indeed their professors – know very little about this research or how to apply it. Weinstein and Sumeracki’s book explains how psychologists generate evidence on learning, gives a basic account of human cognitive processing, explains some strategies for effective learning, and gives tips for applying them. It is not about mathematics and it certainly will not make advanced mathematics simple, but I think that we would all have an easier time if we were more aware of some common misunderstandings about learning and effective ways to improve it.  

For me, this book got the closest to the nitty-gritty of why mathematicians like me, whose job is to prove theorems, find this activity so compelling.

It’s always been the long hunt, with all the frustration as well as the occasional success, that I’ve found so addictive. Doxiadis brought out the nuances of such pursuits brilliantly – the wily Uncle Petros tells the narrator to prove a mathematical statement despite knowing it is almost surely false.

Ah, these little tricks that we mathematicians enjoy playing on unsuspecting souls (I’ve been known to do this to my students a couple of times).

It is fair to say that many people—even those who loved mathematics as students—view mathematics as having always existed. The idea that definitions and theorems that fill our school textbooks were created or discovered by human beings is something that has never crossed their mind. In fact, mathematics has a long, fascinating, and rich history, and William Dunham’s Journey Through Genius is a perfect introduction to the topic. Dunham expertly writes about the history of topics like geometry, number theory, set theory, and calculus in a way that is entertaining, understandable, and rigorous. After finishing Journey Through Genius, readers will not think about mathematics in the same way, and they will be eager to learn about the history of other mathematical topics, people, and cultures.

First published in 1937, this lovely book is a true classic. In two volumes Bell brings to life 30 or so mathematicians, from Archimedes to Cantor. When first reading the book many years ago I had remembered some of the names from school and college, but only as labels to theorems or equations, and I felt taken into a delightful new realm of knowledge – I could now think of Fermat, Lagrange, Gauss, and Riemann as people. And I began to want to know more about the scientists whose names I had heard in school and college. Bell’s book had sparked a lifelong interest.

Frenkel came from the Soviet Union, where discrimination against Jews made it impossible for him to get into Moscow State University. During the oral exam they sent two graduate students to question him, pick holes in his responses, and ensure he failed. He turned to an informal network of Soviet mathematicians for help.

Like him, they were denied serious employment in the field, but after the 'cold war' against the Soviet Union, Harvard invited him to take a fellowship that later turned into a permanent job. Years later, when his old tormentor from Moscow State arrives to give a talk, he confronts the man in a lecture room with first-hand evidence of allegations against the system. Faced with a victim, the Russian mathematician's denials rang hollow. 

This book reaches beyond mathematics to anyone of independent thought in an environment where it is not permitted to step out of line or,… show more.

One way of enlivening any presentation of mathematics is by including some history of the subject, but this only really works if there is some serious scholarship behind it. I especially like Hollingdale's book, partly because of the concise writing style, and partly because of the unusually good balance between history and mathematics itself. The calculus, in all its various forms, with some aspects going right back to the Ancient Greeks, is treated especially well.

John Urschel is an African-American mathematician specializing in graph theory, who recently completed his Ph.D. in mathematics at MIT. But he is better known for his football career, as a starting offensive lineman for the Baltimore Ravens. Six of Urschel’s papers were completed while he was still in the National Football League.

Mind and Matter is John Urschel’s memoir, co-authored with his wife Louisa Thomas. Each chapter alternates between football and mathematics, and how his success on the field translated to success in the classroom, and vice-versa. I loved how accessible the book is, for readers of all ages, and I fully agree with the author’s perspective that mathematics gives us a way of making sense of the world, and helping us see past the confusion of everyday life.