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Bernhard Bolzano was a Czech philosopher, mathematician, and theologian who made significant contributions to both mathematics and the theory of knowledge.
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 entered the Philosophy Faculty of the University of Prague in 1796. 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 and two days later Bolzano was ordained a Roman Catholic priest.
He was appointed (1805) 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 (1819) after pressure from the Austrian government. 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.
Among Bolzano's most important works are
(i) Rein analytischer Beweis (1817), Pure Analytical Proof, which contains an effort to free calculus successfully from the concept of the infinitesimal,
(ii) Wissenschaftslehre (1837), an attempt at a complete theory of science and knowledge,
(iii) Paradoxien des Unendlichen (1851), a study of paradoxes of the infinite. 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. The work (iii) above was only published three years after his death by one of his students. 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.
References elsewhere in this archive:
Tell me about Bolzano's work on set theory and on topology
References:
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