A second edition of Leo Corry's well-known historical study of modern algebra and the idea of a mathematical "structure" is good news, both for historians and for algebraists. This edition is also a little less expensive than the original hardcover, and that is also very good. As is the book.
As Corry explains in his introduction, his initial question was about the origins of category theory. That theory was one attempt to codify the "structural" approach to mathematics and mathematical objects, but there were others, notably by Oysten Ore and by Nicholas Bourbaki. Thinking about these other attempts revealed that the very notion of a "mathematical structure" has changed over time, and in fact has often been deployed with only the vaguest of definitions of what one means by "structure." So the book became an attempt to sort our how algebra became "structural" and what that means.
The first section of the book focuses on a specific part of algebra: the theory of ideals. Corry traces that theory from Dedekind to Noether, and seeks to determine what makes the latter more "structural" than the former. The second sense takes up three attempts to formalize the "modern" approach: by Ore, by Bourbaki, and in category theory.
"I have tried to show," says Corry in the introduction, "that the structural approach to mathematics did not come about simply because it became clear that mathematical concepts can be formulated abstractly." That is an important corrective to our tendency to retroject our ways of thinking into the past. What seems "natural" to us was revolutionary then, and investigating the reasons for the revolution is a worthy enterprise.
As Corry himself admits, his book is definitely not the end of the story. For example, his discussion is strictly limited to mathematics. One could argue that the same move towards abstraction was happening in all sorts of different cultural enterprises, from painting to physics, in the early twentieth century. Can one trace any influence across different cultural enterprises? If so, the "modern" in "modern algebra" suddenly carries a lot more weight, identifying it with a particular cultural moment. Other historians will have to take up this question. Similarly, Corry says little about other parts of modern algebra, and, as he points out in the introduction, the detailed account of the history of category theory that he initially intended to write is not to be found in this book. "Such a detailed account thus remains for a future undertaking," he says. Perhaps we can expect such an account from Corry himself.
Overall, this is a valuable book. Historians know this already. Algebraists might want to take a look too. After all, the basic question here is one our students often ask us: "why should anyone want to think about mathematics this way?" Corry has gone a long way towards providing us with an answer.
Fernando Gouvêa is professor of mathematics at Colby College. He has taught abstract (or modern) algebra more times than he can count, but he still finds it hard to convince students that thinking this way is worthwhile.
Introduction: Structures in Mathematics.- Structures in Algebra: Changing Images.- Richard Dedekind: Number and Ideals.- David Hilbert: Algebra and Axiomatics.- Concrete and Abstracts: Numbers, Polynomials, Rings.- Emmy Noether: Ideals and Structures.- Oystein Ore: Algebraic Structures.- Nicolas Bourbaki: Theory of Structures.- Category Theory: Early Stages.- Categories and Images of Mathematics.