Elements of Mathematics: From Euclid to Gödel

by John Stillwell

Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics--but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits.

From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics.

Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

Genres: Mathematics, Science, History, Nonfiction, Logic

422 pages, Paperback

Published November 7, 2017

About the author

John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University

Reviews

[Charles · Rating 5 out of 5]

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.

[Lucille Nguyen · Rating 4 out of 5]

An astounding read, going through many of the fields of mathematics and providing a quite accessible introductory lesson to logic and its limitations. Could have done with more explanation of some of the lines of equations, and better placement to not interrupt ideas from narrative, but overall a fantastically executed work.

[Beau · Rating 4 out of 5]

This book is quite like a textbook. There is actual mathematics and theorems are proved. I was familiar with most of the material, but it was presented in a very entertaining way. Many proofs very quite slick. Even though there are chapters on arithmetic, computation, algebra, calculus, combinatorics, and probability, I think it is really a logic book. Everything eventually comes back to logic, specifically the ideas of incompleteness, unsolvability, and reverse mathematics. The author's overriding thesis concerns finding the boundary between elementary mathematics and more advanced mathematics. At first, this seemed unimportant and distracting to me, but eventually I saw that the author's aim was to shed light on this topic by means of reverse mathematics, which I find quite fascinating. In fact, I remember that almost 20 years ago Harvey Friedman came to the University of Minnesota where I was studying logic and spoke on the topic. He really made an impression on me.

[Rajesh · Rating 5 out of 5]

Excellent presentation for non-experts describing the boundary between elementary and advanced mathematics through the role played by "infinity".
Includes much historical and philosophical background

[Sam Enright · Rating 5 out of 5]

This book was great. Pitched at exactly the right level I was looking for. Great history and philosophical asides. My only complaint is that the author is obsessed with what precisely counts as 'elementary' vs 'advanced' mathematics, and the distinction seems forced and is not very interesting.

[Mauricio · Rating 5 out of 5]

Good overview of several branches of mathematics, from an elementary point of view.

[Loai Ghoraba · Rating 5 out of 5]

Very well written