This is an uncorrected reprint of a 1995 revision of a book originally published in 1981. The 1995 edition differed from the 1981 original in that it had problem sets at the end of each chapter. According to the authors this was done so that the book could be used as a textbook.

*The Mathematical Experience* is a very interesting read — it provides a highly personal tour through aspects of mathematics, its history, its philosophy, and its relationship with the “real” world. As such it provides a nice glimpse into how two mathematicians thought about their discipline as of some 30 year ago. The chapters are only loosely connected, allowing one to browse without fear.

It is not, in my opinion, suitable as a text. The authors suggest it might be used as either a text for a liberal arts mathematics course or for students preparing to teach secondary mathematics. I think it misses the mark in both instances. Its age alone ensures that many of the details are clearly out of date (the discussion of computers being the most obvious). In addition, it consists of a collection of loosely connected essays which only hang together when one has the necessary context. While senior mathematics majors might make some sense of this, humanities majors enrolled in a course based on this book most certainly would not.

Finally, the Assignments and Problem Sets don’t seem appropriate to me. Many are clearly intended to invoke a serious, reasoned response. While that’s fine in theory, my experience suggests that most students would respond in a perfunctory way to questions such as:

How does mathematics relate to your religion, or to whatever religion you are most familiar with? (page 71)

Choose one theme from this chapter and describe it in a one-page essay. (page 399)

In other cases, problems rely on background knowledge which not all students (certainly not the typical liberal arts student would possess:

Collect evidence to decide which of the following statements are true. Then try to prove them using mathematical induction

a) For every positive integer n, 2 is a factor of n^{2}+n

b) The sum of the odd positive integers from 1 to n is n^{2}

(pages 219–220; induction is not discussed in the chapter)

or

Let S be the set of all true mathematical statements

a) Is this a legitimate set? Discuss

b) How “large” is S?

c) Is your answer to b) a mathematical statement?

(page 353)

That being said, I think this book would make a worthy addition to the libraries of mathematicians interested in the scope and nature of our discipline. Here are a couple of samples.

My favorite section was entitled “The Ideal Mathematician” and consisted in a series of imaginary interviews with a mathematician who specializes in “non-Riemannian hypersquares.” In the course of the interview we learn that mathematicians can seldom talk meaningfully about their research areas with other mathematicians, let alone the literate, non-mathematical public. In view of the increasing dependence of mathematics on outside funding, it’s a concern we all share.

In a section entitled “The Classic Classroom Crisis of Understanding and Pedagogy” we are introduced to the two-pancake problem in the context of a discussion of the difference between a mathematically precise proof and the sort of proof which enables undergraduate (or high school) students to understand the idea behind the theorem. The theorem (page 307) is: the area of two plane pancakes of arbitrary shape can be simultaneously bisected by a single straight-line cut of the knife.

On page 279 we encounter a nice discussion of Fourier series motivated by the physics of music. Unfortunately, there is no discussion of the modern synthesizer, a device most of our students are familiar with. This is one of several sections brought together under the chapter title “Selected Topics in Mathematics.” Others include group theory and the classification of finite simple groups, the prime number theorem, non-euclidean geometry, non-cantorian set theory, and non-standard analysis — all this in a scant 75 pages!

I really enjoyed reading *The Mathematical Experience* and would recommend it for college and personal libraries.

Richard Wilders is Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences and Professor of Mathematics at North Central College. His primary areas of interest are the history and philosophy of mathematics and of science. He has been a member of the Illinois Section of the Mathematical Association of America for 30 years and is a recipient of its Distinguished Service Award. His email address is [email protected].