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Der Briefwechsel Richard Dedekind-Heinrich Weber

Katrin Scheel
Publisher: 
Walter de Gruyter
Publication Date: 
2015
Number of Pages: 
490
Format: 
Hardcover
Series: 
Abhandlungen der Akademie der Wissenschaften in Hamburg 5
Price: 
168.00
ISBN: 
9783110368048
Category: 
Monograph
[Reviewed by
Fernando Q. Gouvêa
, on
05/7/2015
]

Richard Dedekind is an inescapable figure in the history of mathematics in the nineteenth century. It is not just that he proved important theorems, though he certainly did. It is more that he introduced many conceptual moves and ways of thinking about mathematics that laid the groundwork for what Jeremy Gray has described as “mathematical modernism,” using set theory, abstraction, axioms. Later on, Emmy Noether was to remark that “it’s all in Dedekind!” It was, at least in germinal form, though it probably took real genius to see it.

One example of Dedekind’s influence is what I have nicknamed “the Dedekind move”: defining a new mathematical object to be a set that gives sufficient information to capture the properties we want it to have. Dedekind was the first to define an object in a quotient ring as a congruence class, he replaced Kummer’s “ideal divisors” by the set of elements they divide (which is “an ideal”), defined real numbers as certain pairs of sets of rational numbers. Like many of Dedekind’s methods, this has become so pervasive in modern mathematics that we hardly notice it any more.

More relevant to the book under review, Dedekind seems to have been the first to realize that there is a close analogy between a ring of polynomials such as C[x] and the ring of integers Z, which extends to rings of algebraic functions (on one side) and of algebraic integers (on the other). This analogy, which has proved to be incredibly valuable both in number theory and in algebraic geometry, was first developed carefully in a famous paper by Dedekind and Weber, available in English translation as Theory of Algebraic Functions of One Variable. The correspondence collected in this book will clearly be of interest to anyone studying the origins of those ideas.

Dedekind and Weber also worked together on another significant project: the publication of Riemann’s Collected Works. The introduction to this volume divides the correspondence accordingly: material related to Riemann’s works, material related to joint papers, and material related to personal friendship. Everything is beautifully presented: there are brief biographies of both Dedekind and Weber, followed by the letters organized by year, a few letters from other correspondents, and full documentation. Everything is, of course, in German, the original language of the correspondence. The result is an essential book for anyone interested in nineteenth century mathematics.


Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. Dedekind is one of his favorites.