A Brief History of Mathematical Thought

The author tells us in the preface that he wrote this book to “show how the language of mathematics has evolved, and to indicate how mathematical arguments relate to the broader human adventure.” To this end, the book is divided into thirteen chapters, the first twelve of which together “trace out a history of mathematical practice, with a focus on conceptual innovations. I do not claim to have covered all of the key ideas, but I have tried to sketch the major shifts in the popular understanding of maths.” The last chapter is somewhat more philosophical; it uses the material of the previous chapters to address the author’s thoughts about the nature of reasoning.

The first twelve chapters can be conveniently divided into four groups. The first (consisting of chapters 1–4) addresses early mathematics, and discusses the development of number systems, the contributions of the ancient Greeks, and the historical development of algebra. The next four chapters discuss the development of modern mathematics: calculus, non-Euclidean geometry, topology, and changing ideas of infinity, culminating in a discussion of Cantor’s work. Of course, from set theory it is a relatively small step to logic, and the next three chapters are unified by this topic: the author takes us from classical logic through the works of Turing and Gödel. Chapter 12 then discusses the use of mathematical models in science, including not just physics but also biology.

This is, of course, not the only book to organize a history of mathematics around major developments in the subject. The most directly similar book that I know of is Turning Points in the History of Mathematics by Grant and Kleiner. (After all, what is a “turning point” if not a “conceptual innovation” or “major shift”?) Turning Points, like this book, is organized around a series of chapters, each one discussing some major new development in the history of mathematics; it suffers by comparison, however, in that it is much too short — at 100 pages, it looks more like a pamphlet than a book, and does not really develop any idea in any kind of depth. Heaton’s book is about three times as long, and therefore gets to spend more time discussing the various paradigm shifts that are discussed in it. He also provides proofs of some things that are not — but should be — proved in Turning Points, such as the irrationality of \(\sqrt2\).

Yet there are also some puzzling omissions in Heaton’s book. The history of complex numbers is not addressed, and neither is there discussion of cubic or quartic equations. Logarithms and trigonometry are also mostly missing in action. Although the history of calculus is discussed (without, as far as I can tell, ever using the word “fluxion”), the rigorization of calculus in the 19th century is not. (I saw no reference, for example, to Weierstrass anywhere in the text.) Groups are mentioned in connection with symmetry, but there is no real discussion of the historical development of abstract algebra. Also, I think that any serious investigation of the nature and development of mathematical reasoning should spend at least some time discussing the idea of proving things using computers, such as in the proof of the Four Color Theorem. For that matter, the Four Color Theorem is not discussed at all, and neither is that other famous previously-unsolved problem, Fermat’s Last Theorem. (Fermat’s little theorem is mentioned, by that name, but the author does not explain why it is considered “little”.)

These omissions render somewhat problematic the use of this book as a text for a course in the history of mathematics. But I’m not sure that it is really intended to be a textbook; it does not have the trappings of a textbook (there are no exercises, for example) and is part of a series of books that are not textbooks. (The titles in this series, the existence of which I was unaware of until reading this book, all begin “A Brief Guide to…” or “A Brief History of….”, and cover a fascinating and eclectic array of topics, including Sherlock Holmes, James Bond, King Arthur, Jeeves and Wooster, and Bad Medicine. I look forward to making the acquaintance of some of these other entries.)

Another book that most definitely is a text and, though not organized quite as similarly to Heaton’s text as is Turning Points, does have some functional features in common with it, is Math Through the Ages (Second Expanded Edition) by Berlinghoff and Gouvêa. MttA, as I shall henceforth refer to it, begins with an outline of the history of mathematics “in a large nutshell,” filling roughly 65 pages, and then segues into a series of 30 “sketches”, each one discussing some major development of the subject. These sketches are roughly comparable to the discussions in Heaton’s text, and there is some considerable overlap in topics (including the development of various kinds of numbers, algebra, the history of calculus, Euclidean and non-Euclidean geometry, infinity and the theory of sets). MttA, however, also discusses the topics, mentioned earlier, that are omitted in Heaton’s book.

In addition, MttA contains exercises and projects for the student. (There is also an “unexpanded” edition that does not.) It also has, in addition to a bibliography, a very helpful section titled “What to Read Next”, containing not just a list of books but also the authors’ opinions of them. Heaton’s text has a bibliography, but it, too, had some omissions; I can’t imagine a bibliography in a history of mathematics text, for example, that does not mention Katz’s A History of Mathematics, a book that I think probably represents the gold standard among the telephone-book sized mathematics history compendiums that are currently available. (Heaton does mention, however, the mathematics history books by Stillwell, Boyer and Kline.)

MttA, I think, is also a bit more careful with the facts than is Heaton’s book. For example, the author mentions Galois briefly on pages 86–87 and states that although there are formulas for solving cubic and quartic equations, “Galois is famous for proving that an equivalent formula [for solving fifth degree equations] cannot possibly exist”. This seems to me to be a little misleading: in fact, it had been proved before Galois (by Ruffini in an incomplete proof, and about 25 years later by Abel in a mostly complete one) that no general formula for solving a fifth-degree polynomial exists; Galois expanded on these ideas and proved precisely what polynomial equations were so solvable. See, for example, Tignol’s book Galois’ Theory of Algebraic Equations. I did not see either Abel or Ruffini mentioned in the text.

Notwithstanding these concerns, I do not, however, want to give the impression that I think that this is a bad book. I don’t. I think that organizing a book around major events in the history of mathematics is a good idea, and Heaton’s discussions are interesting, generally well-written (though perhaps not as chatty and reader-friendly as is MttA) and offer food for thought.

Even if not used as a text for a course in the history of mathematics, this book would help an instructor organize such a course and also provide useful ideas for classroom lectures, particularly because there is some material here (the philosophical discussions, for example) that seems to me to be not common currency in other books of this sort. So, if I get to teach a course in the history of mathematics again, I will continue to use MttA, which I used successfully a year ago when I last taught the subject, but I will keep this book close at hand for supplemental reading.

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