Bernard Bolzano was a Czech philosopher, mathematician, and theologian who made significant contributions to both mathematics and the theory of knowledge.
Bolzano entered the Philosophy Faculty of the University of Prague in 1796, studying philosophy and mathematics. Bolzano wrote
My special pleasure in mathematics rested therefore particularly on its purely speculative parts, in other words I prized only that part of mathematics which was at the same time philosophy.
In the autumn of 1800 he began 3 years of theological study. While pursuing his theological studies he prepared a doctoral thesis on geometry. He received his doctorate in 1804 writing a thesis giving his view of mathematics, and what constitutes a correct mathematical proof. In the preface he wrote:-
I could not be satisfied with a completely strict proof if it were not derived from concepts which the thesis to be proved contained, but rather made use of some fortuitous, alien, intermediate concept, which is always an erroneous transition to another kind.
Two days after receiving his doctorate Bolzano was ordained a Roman Catholic priest. However, as Russ points out in [33]:-
He came to realise that teaching and not ministering defined his true vocation.
Also in 1804, Bolzano was appointed to the chair of philosophy and religion at the University of Prague. Because of his pacifist beliefs and his concern for economic justice, Bolzano was suspended from his position in 1819 after pressure from the Austrian government. Bolzano had not given up without a fight but once he was suspended on a charge of heresy he was put under house arrest and forbidden to publish.
Although some of his books had to be published outside Austria because of government censorship, he continued to write and to play an important role in the intellectual life of his country.
Bolzano wrote Beyträge zu einer begründeteren Darstellung der Mathematik. Erste Lieferung (1810), the first of an intended series on the foundations of mathematics. Bolzano wrote the second of his series but did not publish it. Instead he decided to
... make myself better known to the learned world by publishing some papers which, by their titles, would be more suited to arouse attention.
Pursuing this strategy he published Der binomische Lehrsatz ... (1816) and Rein analytischer Beweis... (Pure Analytical Proof) (1817), which contain an attempt to free calculus from the concept of the infinitesimal. He is clear in his intention stating in the preface of the first that the work is
a sample of a new way of developing analysis.
Although Bolzano did achieve exactly what he set out to achieve, he did not do this in the short term, his ideas only becoming well known after his death. In [33] Russ describes Bolzano's aims in the 1817 paper:-
In this work ... Bolzano ... did not wish only to purge the concepts of limit, convergence, and derivative of geometrical components and replace them by purely arithmetical concepts. He was aware of a deeper problem: the need to refine and enrich the concept of number itself.
The paper gives a proof of the intermediate value theorem with Bolzano's new approach and in the work he defined what is now called a Cauchy sequence. The concept appears in Cauchy's work four years later but it is unlikely that Cauchy had read Bolzano's work.
After 1817, Bolzano published no further mathematical works for many years. However, in 1837, he published Wissenschaftslehre, an attempt at a complete theory of science and knowledge.
Between sometime before 1830 and the 1840s, Bolzano worked on a major work Grössenlehre. This attempt to put the whole of mathematics on a logical foundation was published in parts, while Bolzano hoped that his students would finish and publish the complete work.
His work on paradoxes Paradoxien des Unendlichen, a study of paradoxes of the infinite, was published in 1851, three years after his death, by one of his students. The word set appears here for the first time. In this work Bolzano gives examples of 1-1 correspondences between the elements of an infinite set and the elements of a proper subset.
Most of Bolzano's works remained in manuscript and did not become noticed and therefore did not influence the development of the subject. Many of his works were not published until 1862 or later.
Bolzano's theories of mathematical infinity anticipated Georg Cantor's theory of infinite sets. It is also remarkable that he gave a function which is nowhere differentiable yet everywhere continuous.