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The Mathematical Works of Bernard Bolzano

Steve Russ
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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Here is cause for celebration. Steve Russ and Oxford University Press have given us an English edition of (a selection of) The Mathematical Works of Bernard Bolzano. What else is there to say but "thank you"?

Despite his fame as one of the first people to really think about the foundations of analysis, much of Bolzano's mathematical works had not appeared in translation until now. Russ's volume contains all of the mathematical works published during Bolzano's lifetime, and also a couple of works that were prepared for publication but not published until much later. It does not include Bolzano's extesive mathematical diary or his work on logic.

Bolzano was a Christian humanist philosopher whose goal, in a sense, was to reorganize and reframe all of human knowledge. His collected works, which are in the process of being published, will include 120 volumes. Bolzano's mathematical work, then, was just a small portion of a much larger project. Russ emphasizes this in his introduction, expressing the hope that his translations may encourage students (and their advisors) to dig into this vast material in order to come to a better understanding of Bolzano's thought.

The works included here are divided into three groups, each of which has a separate introduction by the editor. The first, on "Geometry and Foundations", includes Considerations on Some Objects of Elementary Geometry and Contributions to a Better-Grounded Presentation of Mathematics. These titles already highlight Bolzano's typical concerns: how ideas and objects are to be represented, how knowledge is to be discovered, organized, and presented. The second section, "Early Analysis", contains texts on the binomial theorem, on the intermediate value theorem, and on integration. The final section, "Later Analysis and the Infinite", includes Bolzano's theory of the real numbers, work on functions, and his famous Paradoxes of the Infinite.

The conclusion is clear: no self-respecting library will be without this book. History fanatics may want their own copies. And let's hope that Russ's expectation that his book will stimulate further work will be fulfilled.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College and the editor of MAA Reviews.

Geometry and Foundations
Early Analysis
Later Analysis and the Infinite