Many of the important subjects in 19th century mathematics are very technical and specialized, and most of them (the study of elliptic functions, for example) were "joint projects" of a whole generation of mathematicians. In contrast, set theory is (at least in its simpler parts) a familiar component of every mathematicians tool kit. It is also largely the work of one man, Georg Cantor. Both things make it a particularly inviting topic for historical analysis. This book, which has become a classic, provides a close look at Cantor's work on set theory that pays close attention both to its mathematical and philosophical sides.
There are many stories about Cantor that circulate among mathematicians. Several of them seem to have been created by E. T. Bell in his Men of Mathematics. Like many of the essays in that book, Bell's account of Cantor's life is exciting and dramatic, but mostly false. I'll cite only the most obvious example: Cantor was not Jewish.
Unfortunately, many of Cantor's papers have not survived the wars that shook Europe in the 20th century. As a result, it's not clear that a complete biography of Cantor will ever be written.
In any case, this book is not an attempt at such a biography. The story of Cantor's life is mentioned only as necessary to put the work in context. Dauben is writing not about the life or the man, but about the work. Dauben begins with a careful analysis of Cantor's work on trigonometric series, showing that it raised hard questions about sets of points in the real line. What kind of point sets could exist and how might one describe their properties? Cantor started from there and slowly separated out the notions of cardinality, order type, continuity, limit points, and so on.
Cantor's theory of set points is unusual also in that it generated philosophical debate. Kronecker's opposition, which greatly worried Cantor, was not based on any feeling that the mathematics was incorrect. Instead, Kronecker claimed that while the mathematics might well be correct, it was empty, because the entities about which Cantor wrote simply did not exist. More complex discussions followed, and they involved not only mathematicians, but also philosophers and theologians.
Dauben analyzes all this in detail, giving us a deep understanding of Cantor's progress. For the most part, he adopts Cantor's original language and presents his original arguments. This works well most of the time, but at times it makes it hard to follow what is going on. At certain points, there are what seem to be mathematical errors. It is not clear, however, whose errors they are. Are they typos? Errors made by Cantor? Errors made by Dauben? Or am I missing something? At such points, an analysis of Cantor's argument in modern terms would have helped clarify the situation a lot.
Of course, this is a touchy issue with historians. There is (or was?) a kind of mathematician-turned-historian who tended to silently translate ancient material into modern terms (see for example, P. N. Ruane's review of Witmer's translation of Viète). Sometimes this was done to ridiculous excess. In reaction to this, historians tend to be very careful to stay within the thought world of the period or person under study. In general, the historians are right, but it would be nice if in addition they occasionally gave us an analysis of how one might understand the argument in modern terms. This could have been done at several points in this book in order to clarify the arguments.
Overall, however, one should say that Dauben's study is both useful and insightful. It puts Cantor's work in context, explains how Cantor developed it, gives useful information about the philosophical and theological sides of the conversation, and even gives the reader some hint of how things continued to develop after Cantor. Anyone who wants to understand how modern set theory was born should read this book.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.