The best books on the power and wonder of mathematics

This is the only book about math that has ever caused me to laugh out loud. For Ellenberg, a mathematician at the University of Wisconsin, math is not a set of techniques we learn in school but a commonsensical way of looking at the world. And since we all do it anyway, we might as well learn to do it right, or we will end up very very wrong.

Ever looked at a regular pattern and assumed that it will continue that way indefinitely? Ellenberg suggests you pay attention to that missile, which for the first part of its flight headed straight (and safely) into the sky, but is now headed right towards you in its parabolic trajectory. The lesson? Curves are not always straight lines! Ever wonder why South Dakota has one of the nation’s highest rates of brain cancer, and North Dakota one of the lowest? No, it’s not the malignant effect of Mount Rushmore; it’s the Law of Large Numbers at work. And if you want to know if a statistical effect is significant, you had better listen to what a dead fish with uncanny mindreading abilities has to say.

How Not to Be Wrong is filled with such gems from cover to cover. It is the most fun math book you will ever read, but it is also deep: You will never look at the world the same way again.

Logicomix is an impossible book: a graphic novel about the foundational crisis in mathematics in the early decades of the 20th century. And if that sounds to you crazy, or boring, or contrived (as it did to me when I first heard about it) Doxiadis and Papadimitriou are here to prove you wrong. The book is absolutely beautiful, the artwork exquisite, and most importantly – the story is downright gripping. Once you’ve picked up Logicomix, chances are you will not put it down until you’ve read the whole thing, cover to cover. It’s just that good.

The book follows the dramatic life of Bertrand Russell, and his lifelong quest to achieve absolute certainty. It begins with him as a young boy growing up in his grandparents’ house, trying to understand a world permeated by dark family secrets, covered-up by lies. It ends on the eve of World War 2, with Russell old, famous, and disillusioned, no longer sure that rational unimpeachable truth is possible. Along the way, we meet a cast of colorful characters including the mathematician Georg Cantor, the philosopher Ludwig Wittgenstein, the logician Kurt Gödel, several of the women in Russell’s life, and many others.

Logicomix is a wonderful read, but it is also so much more. Seemingly without effort, it tells the story of a profound crisis that shook mathematics and philosophy, and opened a path to the computerized world we know today.

Love and Math is a mathematical autobiography, seamlessly interweaving an inspiring personal journey with profound mathematical ideas. Born in the Soviet Union, Frenkel aspired to become a professional mathematician, only to see his hopes crushed by entrenched antisemitism at Moscow State University – home to the premier mathematics program in the country. While sitting for the entrance exam, he was confronted by two advanced graduate students who were sent to question him personally and make sure he failed. Rejected but undeterred, Frenkel turned instead to an informal network of top-flight but marginalized Soviet mathematicians, who like him were denied employment in the field they loved. With the end of the Cold War he was invited to Harvard on a fellowship that later turned into a permanent job. In one of the book’s emotional highs, he confronts his old tormentor from Moscow State, who unsuspecting American academics had invited to give a talk.

Frenkel’s love of mathematics oozes from every page of this book. His field is the Langlands Program, an ambitious effort to unify the disparate fields of research in mathematics. Using vivid analogies he conveys to the reader the essence (though not the details) of such constructs as Galois groups, Kac-Moody algebras, and Hitchin moduli spaces. Beneath these seemingly unrelated objects, he believes, lies a single mathematical order, a deep truth unifying them all.

In Love and Math the personal and the mathematical unite. For Frenkel, mathematics is not just an object of fascination, or love. It is also a perfect world of rationality, order, and beauty, a world which the bullies of Moscow State can never touch.

That Steven Strogatz, Cornell Professor and longtime New York Times columnist, is unsurpassed as an expositor of mathematics, goes without saying. No one can make the abstract and technical appear simple and intuitive like Strogatz. In Infinite Powers he takes on the Calculus -- the central pillar of modern mathematics that is also the bane of many a high-school student. It is an immensely powerful field, and at its core is a concept that is both counter-intuitive and paradoxical: the infinite.

The roots of the calculus, we learn, go back to the ancient Greeks, whose notions of the infinite were put to powerful mathematical use by Archimedes. Strogatz continues with Galielo’s dynamics and Kepler’s laws of planetary motion, before reaching the turning point: The discovery of the Calculus by Newton and Leibniz. This leads straight to a discussion of differential equations, which are responsible for so much of what makes our world modern – from cell-phones to jetliners to quantum mechanics.

Strogatz is an applied mathematician, and so his interest is not just with the beauty of math, but in how it helps us understand – and create – things. At every step he provides vivid examples about how the concept he talks about are being used today. Who ever knew that Archimedes’ way of estimating the area of a surface by dividing it into triangles is key to modern computer animation? Or that the calculus was key to developing the drugs that fight AIDS? Strogatz knows all this, and much more. In Infinite Powers he shares it all with the reader.

What is the connection between Category Theory and the social structures of white privilege? Or between Venn diagrams and U.S. immigration policy? If you are like me you probably never thought there was a connection. But in The Art of Logic in an Illogical World Eugenia Cheng shows otherwise. Because when properly applied, she argues, even seemingly esoteric mathematical fields can reveal the hidden logic in the world around us.

Cheng is a research mathematician at the School of the Art Institute of Chicago, and she believes that mathematics has much to teach us about how to understand the world, and how to conduct ourselves within it. Mathematics is a universe ruled by logic alone, where long chains of irrefutable deductions determine all things. There are no arguments in mathematics, only proofs. Our daily lives, in contrast, seem to be made of nothing but arguments, conflicting perspectives, and irrational emotions. According to Chang, however, we don’t need to give in to the chaos: If we only apply the logical methods of math to our human world we will find that it too is pervaded with logic and order. We might find, for example, that some of our seemingly intractable disagreements, are only a cover for fundamental agreement; Or that the reason why we cannot reach consensus is that we are proceeding from different assumption. And so on.

Cheng doesn’t believe that everything in our daily world can be reduced to mathematical logic. Unmathematical “gray areas” and powerful emotions also play an enormous part. But if we only apply a bit of mathematical thinking tour culture and politics, we could save ourselves a great deal of time, and a lot of heartache.