Bernard Bolzano, born in Prague in 1781, was a Bohemian mathematician with Italian origins. Bolzano made several profound advances in mathematics that were not well publicized. As a result, his mathematical work was overlooked, often for many decades after his death. For example, his construction of a function that is continuous on an interval but nowhere differentiable, did not become known. Thus, the credit still goes to Karl Weierstrass, who found such a function about 30 years later. Boyer and Merzbach described Bolzano as “a voice crying in the wilderness,” since so many of his results had to be rediscovered by other workers.

Bolzano studied philosophy, physics and mathematics at Charles University. He also studied theology, and was ordained a priest in 1804, just after receiving a doctorate. His thesis was on geometry. Bolzano was greatly concerned with the philosophy of mathematics. In particular, he aimed to put mathematics on a firm logical foundation, and he recognized that many results that were thought to be “obvious” actually required to be proved in a rigorous way. One example is his formulation of what is now called the Jordan Curve Theorem.

**Conflict with Authority**

In 1807 Bolzano was appointed to the chair of the philosophy of religion. In 1818, he was elected Dean of the Faculty of Philosophy at Charles University. He was expected to promote orthodox religious and political doctrines. However, Bolzano had liberal and progressive social views and these brought him into conflict with the religious and political authorities. The rulers of Bohemia in Vienna regarded Bolzano as a trouble-maker, and his theological views were frowned on by the church authorities. He was prohibited from publishing for some years and, as a result, much of his work became known only many years after his death. In 1825, having been tried by the Church, Bolzano was required to recant his expressed opinions, regarded as heretical. He refused to do so and resigned his chair.

**Making Calculus Rigorous**

Ever since its inception, calculus had been plagued by misunderstandings relating to the concept of infinitesimal quantities. The polemic of Bishop George Berkeley, *The Analyst*, is well known. Bolzano made a serious attrempt to free calculus from the concept of infinitesimals. He constructed definitions of limits, convergence and derivatives using what was essentially the epsilon-delta approach used to effect by Weierstrass and still in vogue today.

There is a great similarity between Bolzano’s work in calculus and that of Augustin Cauchy, and between their definitions of limits, continuity, convergence and derivatives. Neither was aware of the work of the other, but it is Cauchy who is credited with the discoveries, as his work was immediately recognized and appreciated, while Bolzano’s work escaped notice. Cauchy proved a necessary and sufficient condition that an infinite series S_{n} converges: for a given positive number ε, however small, there is some N such that all terms beyond S_{N} differ from S_{N} by less than ε. What is today known as Cauchy’s criterion was actually first formulated by Bolzano.

**Analysis**

The Intermediate Value Theorem states that a continuous function defined on an interval [a, b] and having opposite signs at the end-points, must be zero somewhere within the interval. This is a typical example of a result that is “obvious” but that Bolzano felt must be demonstrated rigorously. He gave a proof that was as sound as was possible, given the methods and theory available at the time.

Bolzano was sharply aware of the need to refine the concept of real numbers. His name lives on in the Bolzano-Weierstrass Theorem, a fundamental result in the theory of real numbers. The theorem states that every bounded sequence contains a convergent subsequence. This was first proved by Bolzano in 1817 as a lemma in his proof of the intermediate value theorem. About fifty years later, it was proved again by Weierstrass. It is one of the essential theorems of analysis.

**Paradoxical Properties of Infinite Sets**

Bolzano wrote a book, *Paradoxes of the Infinite,* in which he anticipated several of the issues later examined by Cantor. He gave examples of a one-to-one correspondence between an infinite set and one of its proper subsets, and realized that this was a characteristic property of infinite sets in general. Much earlier, Galileo had noted a one-to-one correspondence between the natural numbers and the perfect squares, n -> n^{2}. Bolzano showed that correspondences between infinite sets and proper subset are common. For example, the simple linear equation y = 2x establishes a one-to-one correspondence between the real interval [0,1] and [0,2]. Arguably, there are as many real numbers in the the intervals, although one is twice the length of the other.

Bolzano also saw that the infinity of real numbers is of a different type to the infinity of integers. In this he anticipated Cantor by a half century. Much later, Dedekind identified the defining characteristic of infinite sets: a set S is infinite when it is in one-one correspondence to a proper part of itself. Dedekind recognized the universal property of infinite sets that he took as a precise definition. But Bolzano had the same idea much earlier.

**Bolzano’s**** Gesamptausgabe**

Bolzano was visionary, and his work adumbrated that of several other mathematicians. As already observed, he gave a definition of what we now call a Cauchy sequence that pre-dated Cauchy’s *Cours d’analyse* by some years. He also constructed a function that was continuous but nowhere differentiable on a closed interval. This was well in advance of Weierstrass’s construction of a function with similar properties.

Although work published in his lifetime was limited, Bolzano was prolific and left a vast corpus of work. His *Complete Works* (*Gesamptausgabe*), now close to completion, contain 102 volumes published and 29 volumes still in preparation. Bernard Bolzano was far ahead of his time, and his mathematical writings and ideas are still of interest today.

**References**

Bernard Bolzano (1781–1848) is “one of the major figures in the history of ideas.” For details of the publication of his Gesamtausgabe, see here: __https://www.frommann-holzboog.de/editionen/20__

Merzbach, Uta C. and Carl B. Boyer, 2011: *A History of Mathematics.* Third Edn. John Wiley & Sons, Inc. ISBN 978-0-470-52548-7.

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