The Mathematical Works of Bernard Bolzano

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Bernard Bolzano (1781-1848, Prague) was a remarkable thinker and reformer far ahead of his time in many areas, including philosophy, theology, ethics, politics, logic, and mathematics.
Aimed at historians and philosophers of both mathematics and logic, and research students in those fields, this volume contains English translations, in most cases for the first time, of many of Bolzano's most significant mathematical writings.
These are the primary sources for many of his celebrated insights and anticipations, including: clear topological definitions of various geometric extensions; an effective statement and use of the Cauchy convergence criterion before it appears in Cauchy's work; proofs of the binomial theorem and the intermediate value theorem that are more general and rigorous than previous ones; an impressive theory of measurable numbers (a version of real numbers), a theory of functions including the
construction of a continuous, non-differentiable function (around 1830); and his tantalising conceptual struggles over the possible relationships between infinite collections.
Bolzano identified an objective and semantic connection between truths, his so-called 'ground-consequence' relation that imposed a structure on mathematical theories and reflected careful conceptual analysis. This was part of his highly original philosophy of mathematics that appears to be inseparable from his extraordinarily fruitful practical development of mathematics in ways that remain far from being properly understood, and may still be of relevance today.

The Mathematical Works of Bernard Bolzano Review

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.

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