Elements of Mathematics: From Euclid to Gödel, by John Stillwell, Princeton University Press, Princeton, New Jersey, 2016. 440 pp., $39.9 (hardbound). ISBN 978-0-691-17168-5.
Five out of five stars
For those experienced in mathematics, the most interesting feature of this book is the attempt to keep things at the elementary level. As is demonstrated in many ways and emphasized by Stillwell, the phrase “elementary mathematics” is one subject to a wide variety of interpretations. Both over time as well as from person to person. What was considered advanced when introduced has become routine over centuries.
Yet, Stillwell does point to one concept that can be used to separate the elementary from the advanced, and that is the idea of infinity. It is an idea that will always remain abstract and requires thought processes that can accept what appears to be paradoxical. For example, the idea that the infinity of the even integers can be contained in the integers and yet they can be considered to be the same “size.” Or that the infinity of the real numbers is “larger” than that of the natural numbers.
Since the coverage begins with Euclid and flows through the centuries until the twentieth, this book is first and foremost a popular history of mathematics. Yet, there is an important underlying theme, that there is a concept that can be used to determine the difference between elementary and advanced mathematics. While it is of course not completely effective as a separator, it is a good first approximation.
This is a book that would serve well as a textbook for a liberal arts course in mathematics. There is no sparing of the formulas, some sections would have to be skipped or subject to deep explanations, yet the coverage of the fundamentals of mathematics is sufficient to justify its use.
This book was made available for free for review purposes.