Elements of Mathematics: From Euclid to Gödel

Acquiring a new book by John Stillwell is always a pleasant experience. Stillwell is, to my mind, one of the better current mathematical authors: he writes clearly and engagingly, and makes more of an effort than most to provide historical detail and a sense of how various mathematical ideas tie in with one another. I have found, over the years, that he can usually be counted on, even when discussing areas that I was already reasonably familiar with, to make some insightful comment that I hadn’t thought about before.

Judging from his last few books, Stillwell seems to have been thinking about infinity a lot recently. A prominent theme of his Yearning for the Impossible is that much of mathematics arose historically because of man’s attempt to grapple with the “impossible” — broadly defined to mean ideas that were so counter-intuitive as to seem impossible. Infinity clearly plays a role in this idea. Subsequently, in Roads to Infinity, a book that Stillwell says in the preface can be viewed as a sequel to Yearning, infinity also plays a major role: the very first sentence in the preface is that “[m]athematics and science as we know them are very much the result of trying to grasp infinity with our finite minds.” The book then proceeds to explore the role that infinity plays in set theory and logic.

After Roads to Infinity, there is The Real Numbers: An Introduction to Set Theory and Analysis; the title itself here makes clear here that infinity and infinite processes figure prominently. And, more recently still, Stillwell has contributed a chapter (“From the Continuum to Large Cardinals”) to Mathematics, Substance and Surmise, a book of essays that was recently reviewed in this column; this essay also is very much involved with notions of infinity.

Infinity also plays a big role in the book under review as well. The basic idea of this book is to discuss eight areas of mathematics that are generally viewed as “elementary”, and also to explore just what it means to characterize something as “elementary” rather than “advanced”. This distinction involves looking at ideas of infinity: “Infinity may not be the only characteristic that defines advanced mathematics, but it is probably the most important, and the one we understand the best.” (Other characteristics that define advanced mathematics, and that Stillwell discusses in the text, are abstraction and proof.)

The eight particular areas selected by Stillwell are, in order of discussion: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability and logic. The first chapter of the book provides a quick overview of each of these topics, each lasting about five pages, give or take. There follows eight additional chapters in which these topics are discussed in more detail, one topic per chapter but with some emphasis on how they relate to one another. A final chapter is on topics that Stillwell views as “advanced”, one topic for each of these eight areas. (Examples: for calculus, Stillwell states and proves Wallis’s product formula for \(\pi\); for algebra, he gives a proof of the Fundamental Theorem of Algebra; for geometry, the projective line and its relationship to linear fractional transformations are discussed.)

The features that we have learned to expect from Stillwell (including, but not limited to, excellent writing) are present in this book as well. Each chapter, for example, contains a section with historical notes; in addition, each chapter contains a section containing philosophical remarks, mostly addressing Stillwell’s thoughts on how to distinguish elementary from advanced results. Some of his observations here provide interesting food for thought, with implications for the teaching of undergraduate mathematics. He believes, for example, that in the study of algebra, rings and fields belong in the “elementary” category, but groups do not. He also believes that non-Euclidean geometry is advanced, but Euclidean geometry is elementary; I wonder, though, if this may have something to do with the fact that most people grow up learning Euclidean geometry, and therefore have a “Euclidean intuition” that they must overcome when studying the non-Euclidean theory.

One of the nice things about this book is that it can appeal to various constituencies at different levels of mathematical sophistication. Students who have just started college and anticipate being mathematics majors can learn quite a bit about what they will be exposed to by reading selected portions of this book, even if not everything they read is completely comprehensible to them at this level. On the other hand, mathematics majors about to graduate from college will undoubtedly find something in this book that they have not encountered before in their studies. The same could actually be said of faculty members, many of whom have likely never actually given any serious thought to what makes a result “advanced” rather than “elementary”, and who may learn something new about mathematics itself. I had never heard of “reverse mathematics”, for example, before reading this book. (For readers of this column who have also not heard of this, it’s a new area of mathematical logic that, in very broad terms, studies theorems by looking at the axioms that are needed to prove them; the stronger the axioms, the more advanced the theorems.)

Although this book does not have the formal trappings of a text (there are no exercises, for example), it could, I think, be used with profit as a text for a capstone course or senior seminar. There is an extensive bibliography for those who might want to explore further. Careful attention paid to this book will reward a student with a deeper understanding of many topics that he or she has studied before, as well as, no doubt, a first look at a number of new topics and ideas that the student has not seen before, but should have. This is a very nice book.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.