Customer Reviews

The Mathematical Works of Bernard Bolzano

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Had Bolzano published more of his mathematics, his reputation today would be much greater. As it was, much of what he did was independently reproduced by later mathematicians and he is not placed as highly in the mathematical pantheon as he deserves. This book, which belongs in the library of any college with a pretense of being a university and on the shelf of any mathematician with an interest in the history of the field, is a real eye opener. In the past few years I have become interested, through the works of van Rootselaar, Gericke, and Laugwitz, in Bolzano's foundations of the real numbers, a work left unfinished and marred by errors which gave me the impression that Bolzano understood the foundational problems better than any of his contemporaries, but was too weak technically to carry out the necessary work. Thanks to this book I have revised my opinion. True, he made some errors--like everyone else he did not realise the necessity of uniform convergence for various results, and in his foundations of the real numbers he didn't think things through enough to complete the project. However, he was heading in the right direction and his work on functions, with the construction of a nowhere differentiable function reveals great technical skill.

I would have liked more annotations and explanatory material. Knowing before reading which parts were in error would be a time saver. And I would have thought any Bolzano fan would have relished in debunking Hahn's claim that the proof of the convergence of the geometric series in Paradoxes of the Infinite was "not grounded on anything". These minor faults aside this book is one of my most valued recent acquisitions.