# Bernard Bolzano: His Life and Work

Bernard Bolzano (1781-1848) is well known among mathematicians. He was however a philosopher and logician in the first place. As a product of the Enlightenment, he had very modern, almost revolutionary, ideas about science, church, and society. This was however not appreciated by the ruling political upper class, since these ideas endangered their absolute power over people.

The first author of this book is specialised in the philosophy of mathematics and the second is a philosopher, both ardently interested in Bolzano and his work. They started working together at the end of the 1990's, which resulted in this massive book project. They could complete it because the majority of the work of Bolzano became available as it is compiled in *Bernard Bolzano: Gesamtausgabe* (Fromann-Holzboog, 1969ff), which is still an ongoing project since 102 volumes are currently (2019) published, and 29 are still in preparation.

Bernard Bolzano was born in Prague in Bohemia, which the authors want to explicitly distinguish from (modern) Czechia, and not only for geographical reasons. More important in this context is its political history. The fact that this region was part of the Holy Roman Empire, and later of the Austrian Empire with oppressive rulers resulted in a smoldering anti-catholic and anti-German sentiment. By the end of the 18th century, the Czech language and culture experienced a revival as a consequence of widespread romantic nationalism, and Bolzano was an exponent of this movement.

Bolzano was raised in a pious catholic family (12 children but only two reached adulthood). His health has been weak throughout his life. Although his father wanted him to become a merchant like he was himself, Bolzano started studying mathematics and theology and became a catholic priest in 1804. He started teaching the philosophy of religion at the Charles University in Prague where he became a popular teacher, loved by his students. However, Bolzano's rather liberal opinions about the church and about politics, made him not popular among his superiors. The Austrian government however was suspicious of his ideas spreading among his students. They put some pressure on the local authorities, which led to Bolzano's suspension as a professor in 1819 and he was put under house arrest. Later he was also tried by the church and, since he refused to recant his "heresy", he retired from his chair and spent the summer with his friends the Hoffmanns outside Prague, and only returned during the winter period. This gave him ample time to work on his mathematical and philosophical texts. However many of his writings had only limited distribution because of his conviction official publication was almost impossible. Other texts remained unpublished until 1962 or even much later. This explains why he had relatively little direct impact and some of his original ideas were later rediscovered by others.

Among his major publications were his Rein analytischer Beweis. Here he tried to remove infinity from calculus, and hence had to define limit, continuity, derivative, and convergence without it. In this context we find his treatment of the intermediate value theorem and a definition of a Cauchy sequence. So he developed these ideas some years before Cauchy. He used the Bolzano-Weierstrass theorem many years before Weierstrass did. His Grossenlehre is his attempt to start setting up a logical foundation of mathematics, which he generalized in his Wissenschaftslehre to a complete theory of knowledge. In his Paradoxien des Unendlichen (Paradoxes of the infinite) the word "set" is used and he also has the bijection between the elements of an infinite set and an infinite subset. Many other results stayed unpublished for a long while until in the 20th century, and hence are attributed to other mathematicians. For example in his discussion of continuous functions, he had some fractal-like sawtooth monster-function, that became known as a Weierstrass function, and there are several other examples discussed in the book.

Those strict mathematical subjects in this book are discussed in a relatively short chapter, but the other aspects of Bolzano's work, mostly philosophical, are even more thoroughly discussed in the other chapters. That includes his opinions about ethics, political philosophy, philosophy of religion and the catholic church, about aesthetics and a science of beauty, as well as about ontology and metaphysics. A large chapter is devoted to logic and another one to his Wissenschaftslehre (Theory of science). Concerning logic, Bolzano was of the same idea as Leibniz who was convinced that logic had an important part in the philosophy of science, and he is at the origin of the interaction between logic and mathematics. This was opposed to Kant, who thought there was no role for logic in philosophy, and that what was used in mathematics was a completely different thing. Logic became a core aspect of Bolzano's philosophical work and it is a hidden precursor of what later became known as analytical philosophy, for which Gottlog Frege is usually considered to be the founding father.

His Theory of Knowledge, just like his logic, starts from the concept of truth by which he means the "truth in itself". Several such propositions do exist outside our mind. We do not have to reason about these. This sounds Cartesian, but he considered it not to be fundamental but as a way to refute scepticism. Then these propositions are brought in relations, inductions, etc., which is the logic in a narrow sense. Only after eleven hundred pages he comes to the theory of knowledge: ideas that can be conceptual or empirical and they are subject to judgement. New propositions can be obtained by logic, probability, or what is called an entailment relation (a set of propositions can entail a new proposition).

This book contains a thorough analysis of the philosophical work of Bernard Bolzano. Much of his work, has for a long time been unpublished, so that this book comes at a good time with the complete and very extensive scientific and philosophical production of Bolzano becoming more generally available in the Gesamtausgabe. The discussion of his mathematics and of his ideas on all aspects of science and society, are strictly documented with many citations from the original sources (in English translation) and from other authors that have studied his work. It is also placed in relation with other philosophers and mathematicians. This book is in the first place about the philosophy (of mathematics) as it can be found in Bolzano's writings. The strict mathematics in a narrow sense are omnipresent but at a second level.

**Submitted by Adhemar Bultheel |

**3 / Jun / 2019