The Foundations of Mathematics

“What is mathematics?” I think this is a necessary question that every serious student of mathematics and their instructors should confront, even perhaps on several different occasions over a period of time. Do our answers change? I would certainly hope so! A solid course of studies in the foundations of mathematics should help to clarify, if not partially answer, such a question.

Over the years, when I have taught a course in the foundations, I always begin and conclude the course with this question, seeking to gauge the changing mathematical maturity of my students. The foundations course is the “boot camp” for anyone intending to participate in mathematics beyond the elementary level. It establishes a framework of better understanding for the mathematical concepts and structures that will be encountered in the ensuing years. I believe it is one of, if not the most important, subject of study for a person intending to explore higher mathematics. In such a learning situation, a good textbook is vital.

One of the most recent contributions to this effort is The Foundations of Mathematics by Ian Stewart and David Tall. This is a revised and expanded, second edition, of their 1977 book. I have not seen the previous book, so my comments and evaluations are limited to this edition. Both Stewart and Tall, now retired professors of mathematics, bring most appropriate backgrounds to the writing of such a book. Ian Stewart is a popularizer of science and mathematics, the author of numerous informative and engaging works on those subjects. David Tall is also an accomplished author, a researcher and mathematics educator whose most recent efforts have focused on the subject of mathematical thinking. Both these men bring their wealth of experience and talent to shaping the content and instructional direction of this text.

The book contains all the usual topics one would expect to find in a foundations text: naïve set theory, relations, orderings, functions and operations; an introduction to elementary logic and mathematical proof. Beginning with the Peano’s axioms, the real numbers are constructed and further defined as a complete ordered field. An innovative feature is the consideration of infinitesimals as inverses of a subfield of the positive real numbers. Cardinal and transfinite counting and arithmetic are also considered.

A feature that I particularly appreciate is that instead of merely outlining formal mathematical systems, the authors introduce the readers to the concepts and structures of group, ring, ideal, fields and vector spaces. All subject matter is reinforced with appropriate and understandable illustrations and examples. Readers are provided with the necessary mathematical and historical perspectives to “make sense of” what they are learning .as well as insights into the processes of mathematical thinking.

Pedagogical motivation is evident throughout the text. For example: Chapter 13 focuses on the concept of a group as arising from the mathematics of permutations and the appeal of visual symmetry. Along the way, the reader is introduced to the work of Évariste Galois, Niels Henrik Abel and Felix Klein with his Erlanger Programme — very nice!

In reviewing a book, I usually inspect the Index to see if all the “relevant players” are included. In this instance, most are referenced with the exception of the Bourbaki. I would also have liked to have seen a brief section devoted to the attempts, in the late nineteenth and early twentieth centuries. to place mathematics within the philosophical confines of Logicism, Intuitionism, or Formalism.

In their writing of The Foundations of Mathematics, the authors motivate, encourage and teach their readers. If I were to offer a future course in foundations, I would certainly use this book and highly recommend it to others, both as a reference and as a text.

One word of caution: this text reflects the British teaching experience and as such is probably compiled to suit the needs of “A level” students, advanced students who seek to pursue further studies in science and mathematics. Therefore, instructors wishing to employ this text in “the States” should proceed with some moderation. This book contains enough material for a two-semester approach to the subject. The “tree of mathematics” depicted on the book’s cover with its supporting trunk and roots representing the foundations and its branches and leaves the diverse fields of mathematics, presents a thought-full analogy of mathematics. So too, a reader of this book will attain a clearer perception of mathematics.

Frank Swetz, Professor Emeritus of Mathematics and Education at the Pennsylvania State University, is now occupied as a “Treasure Hunter” for the MAA e-journal Convergence. He collects and annotates historical mathematical materials for the journal’s archive of “Mathematical Treasures”.