# Gallery of the Infinite

###### Richard Evan Schwartz
Publisher:
American Mathematical Society
Publication Date:
2016
Number of Pages:
187
Format:
Paperback
Price:
29.00
ISBN:
9781470425579
Category:
General
[Reviewed by
Megan Patnott
, on
06/9/2017
]

Schwartz’s goal in Gallery of the Infinite is to introduce the reader to the standard introductory material on cardinality, plus a bit extra, beginning with “what is a set.” Although this is popular material in mathematics aimed at the general reader, Schwartz’s take is distinctive in its genuine lack of prerequisites and use of illustrations. The illustrations are an essential part of the book. They use several recurring themes (e.g., an art gallery and infinite animals/people) to assist the written text in explaining abstract ideas, such as the definitions of the counting numbers as sets, with minimal formal notation.

The book has four sections. The first section starts by making a distinction between infinity as the point at the end of the number line and the way infinity will be discussed in the book. It then introduces sets, bijections as a tool for comparing the size of two sets, and the counting numbers as sets. The second section builds on the first by introducing $\aleph_0$ as the set of all counting numbers, and then plays with countably infinite sets.

Cantor’s diagonal argument is the star of the third section. This section begins by using it to show that the set of all binary strings is not countable, and, after introducing the reader to other sets with cardinality $2^{\aleph_0}$ and giving an informal explanation of the Cantor-Bernstein Theorem, ends by returning to Cantor’s diagonal argument in order to show that a set’s power set has strictly greater cardinality than the original set and thus the discovery that there are infinitely many sizes of infinity. The book concludes, in its final section, with a brief discussion of mathematical axioms and the potential problems hidden therein.

Not long after I received my review copy of Gallery of the Infinite, different sizes of infinity came up in a conversation with my friend Alyse, a poet and professor in our English department. Alyse has no mathematical training beyond high school, but she has recently become fascinated with several mathematical concepts related to infinity. I lent her the book, expecting to get it back in a week or two. She kept and studied it the entire semester! She’d gotten stuck a couple of times before stalling on Cantor’s diagonal argument, but she also was intrigued enough to seek out additional explanations and material. The illustrations were helpful, and she enjoyed the poetic phrasing that Schwartz sprinkled throughout the book.

I would cautiously recommend this book. The illustration style won’t be to everyone’s taste (it took me about half the book to get used to it), and the book starts coming apart quickly due to poor binding. It won’t magically give understanding of infinite cardinality to the reader. However, its unique presentation and conversational style makes it a good conversation starter between mathematicians and inquisitive folk with little formal mathematical training, and I can see it capturing the imagination of interested readers from many backgrounds.

Megan Patnott is an Assistant Professor of Mathematics at Regis University in Denver, CO. Her training is in algebraic geometry and commutative algebra.