Beyond Infinity:An Expedition to the Outer Limits of Mathematics

Beyond Infinity: An exploration of the outer limits of mathematics is a book in two parts. The first part, Eugenia Cheng, in her very characteristic narrative, takes us through a variety of mathematical explorations of the idea of infinity and provides us with a beautifully crafted insight into the inner workings of the journey of mathematical thinking. As we turn the pages, we buildup, step by step, a deeper and deeper understanding of what infinity is and what it is not. Cheng takes our mind on a journey of explorations of mathematical ideas and the creation of mathematical objects.  She makes us play with these objects to gain understanding about infinity and as the story unravels we go along with fun mental twists and turns.

What would happen if infinity were a number? She introduces natural numbers, rational numbers, real numbers, all through beautiful analogies and leads us to the discovery that infinity cannot be any of those. So, we conclude, infinity is not a “normal type” of number. So what is it? In the mental mystery of discovering things about infinity we encounter Hilbert Hotels, Cantor’s diagonalization argument, different sizes of infinities, as well as cardinal and ordinal numbers. Cheng excels in her use of analogies and stories to bring her readers closer and closer to the idea of infinity.

Once we have a sense for infinity, the second half of the book illustrates some encounters with infinity of the very big and the very small types. We compare growth rates thinking about folding puff pastry and later through song shuffles. We’re introduced to the idea of dimension, transition into infinite dimensions, and then get a glimpse of category theory. Later we turn to the infinitesimally small and consider infinite sums of smaller and smaller numbers that lead to mind-blowing paradoxes. We get to think about the idea of limits as targets that get smaller and smaller. She fills gaps in the real line and shows that irrationals are indeed on the real line which gets us close to the definition of real numbers. This part of the book is called the sights, and that is exactly how it feels, we get a little taste of infinity in different contexts, and are left wanting more.

Think this book is accessible to anyone curious about infinity. I have found myself recommending this book to my college students as a nice introduction to the idea of infinity, but also as a good introduction to how mathematicians can make up and play with mathematical objects, to build mathematics. It is a fun and instructive reading.

Zdeňka Guadarrama is a Professor of Mathematics, Chair of the Department of Decisions and Mathematical Sciences at Rockhurst University in Kansas City, MO, and Co-Director of Mathapalooza a mathematics community outreach initiative. She is passionate about mathematics education and outreach, with her work currently focusing on curriculum development through inquiry, and the intersections of mathematics with other fields, particularly the arts. She coauthored the book Calculus: A guided Inquiry.