**Bernard Bolzano** successfully freed calculus from the concept of the infinitesimal. He also gave examples of 1-1 correspondences between the elements of an infinite set and the elements of a proper subset.

- Bolzano's upbringing was a major factor in the ideas that he taught later in his life.
- It is fair to say that Bolzano left this environment more convinced of the moral beliefs, which had been foremost in his upbringing and in his schooling, than in the purely religious Christian beliefs.
- Bolzano entered the Philosophy Faculty of the Charles University of Prague in 1796, studying philosophy, physics and mathematics.
- Bolzano was particularly influenced in his mathematical studies by reading Kaestner's Mathematische Anfangsgründe Ⓣ(Mathematical Foundations).
- During the year 1799-1800 Bolzano undertook research in mathematics with František Josef Gerstner and also contemplated his future.
- Two days after receiving his doctorate Bolzano was ordained a Roman Catholic priest.
- However, his professor at the Charles University had put forward an argument which Bolzano had found very persuasive, namely that faith in a doctrine was justified if it led to moral good.
- This allowed Bolzano to accept the mystical elements of Christianity for the greater good of mankind, although he did not accept them to be historically true.
- In order to understand the events of Bolzano's life, we need to fill in a little background about the situation in Bohemia.
- Chairs in the universities were filled by competition and Bolzano entered two such competitions for chairs at the Charles University.
- Bolzano came top in both competitions, but the university preferred to give him the chair in the philosophy of religion since they were then able to give the mathematics chair to Ladislaw Jandera who had substituted for Vydra during his illness between 1801 and 1804.
- In many ways Bolzano was exactly the wrong person to fill this chair given the reasons for its creation, for he stood for all the ideas which Franz feared, being a free thinker who believed in social justice, pacifism, and equality for the Czech speaking Bohemians.
- The appointment of Bolzano was viewed with suspicion by the Austrian rulers in Vienna.
- Some members of the Roman Catholic Church were also unhappy because Bolzano's lectures contains elements of rationalism.
- In 1815 Bolzano was elected to the Royal Bohemian Society of Sciences which was bilingual society drawing its members mainly from the German speakers but also from Czech speakers.
- Bolzano published On the Condition of the Two Nationalities in Bohemia in 1816 in which he put into print his concerns that the Czech Bohemians were dominated by the German speaking Bohemians.
- The peasants were Czech speaking, the cities largely inhabited by German speakers but Bolzano saw the problems which were being created due to increasing industrialisation which saw Czech speakers moving into the cities.
- Bolzano's career continued to flourish, despite the fact that charges were brought against him at the Vienna court in 1816, and in 1818 he was elected Dean of the Faculty of Philosophy at Charles University.
- Bolzano was suspended from his position in December 1819 after pressure from the Austrian government.
- When Anna Hoffmann took ill in 1841, Bolzano and the Hoffmanns moved to Prague where they all lived with Johann Bolzano (Anna died in 1842).
- There Bolzano again became an active member of the Royal Bohemian Society of Sciences and was president during 1842-43.
- Bolzano wrote Beyträge zu einer begründeteren Darstellung der Mathematik.
- Bolzano wrote the second of his series but did not publish it.
- Although Bolzano did achieve exactly what he set out to achieve, he did not do so in the short term, his ideas only becoming well known after his death.
- 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.
- Between sometime before 1830 and the 1840s, Bolzano worked on a major work Grössenlehre Ⓣ(The greater theory).
- This attempted 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.
- 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.
- Bolzano's theories of mathematical infinity anticipated Georg Cantor's theory of infinite sets.
- Attempts to publish Bolzano's manuscripts are described in our article Bernard Bolzano's manuscripts.
- As an example of the mathematics that Bolzano was working on while he was professor of the philosophy of religion, here is a description of what he recorded in his notebook Miscellanea mathematica in 1816.
- In addition to his mathematical work, Bolzano was important as a philosopher and as a logician.
- In a critical exposition, Bolzano presents views of his predecessors and compares them with his own point of view.

Born 5 October 1781, Prague, Bohemia, Austrian Habsburg domain (now Czech Republic). Died 18 December 1848, Prague, Bohemia (now Czech Republic).

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Algebra, Analysis, Origin Czech Republic, Set Theory, Special Numbers And Numerals, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive