Bernard Bolzano

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Bernard Placidus Johann Nepomuk Bolzano (October 5, 1781 – December 18, 1848) was a Czech mathematician, theologian, philosopher and logician. He was born in Prague.

Bolzano entered the University of Prague in 1796 and studied mathematics, philosophy and physics. Starting in 1800, he also began studying theology, became a Catholic priest and was appointed to the chair of religion in 1805. He proved to be a popular lecturer not just in religion but also philosophy, and was elected head of the philosophy department in 1818. However, his political convictions (which he was inclined to share with others with some frequency) eventually proved to be too liberal for the conservative institution, and in 1819 he was dismissed from his positions and exiled to the countryside for the remainder of his life.

Although forbidden to publish in mainstream journals as a condition of his exile, Bolzano continued to develop his ideas and publish them either on his own or in obscure Eastern European journals. Bolzano's early work Paradoxien des Unendlichen (The Paradoxes of the Infinite) was greatly admired by many of the eminent logicians of the day, including Charles Peirce, Georg Cantor, and Richard Dedekind. Despite such ground-breaking contributions to the foundations of mathematical analysis as the introduction of a fully rigorous ε-δ definition of a mathematical limit and the first purely analytic proof of the Intermediate Value Theorem (also known as Bolzano's theorem), much of Bolzano's work remained virtually unknown until Otto Stolz rediscovered many of his lost journal articles and republished them in 1881.

Today he is mostly remembered for the Bolzano-Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano's first proof and which was initially called the Weierstrass theorem until historians of mathematics uncovered Bolzano's earlier work.

In his philosophy, Bolzano developed an ontology in which the world consisted of actual and non-actual objects. Actual objects were further divided into substances such as tables or human beings and the adherents to substances such as colors or mental states. Non-actual objects consisted of non-material things such as numbers and what Bolzano called "Sätze-an-sich" ("Ideas-as-such"). The Sätze-an-sich included what are essentially logical axioms and abstract truths, which Bolzano believed to exist independently of the human mind.

In his 1837 "Theory of Science" he attempted to provide logical foundations for all sciences, building on abstractions like part-relation, abstract objects, attributes, sentence-shapes, ideas-as-such, propositions, sums and sets, collections, substances, adherences, subjective ideas, judgments, and sentence-occurrences. These attempts were basically an extension of his earlier thoughts in the philosophy of mathematics, for example his 1810 Beiträge where he emphasized the distinction between the objective relationship between logical consequences and our subjective recognition of these connections. For Bolzano, it was not enough that we merely have confirmation of natural or mathematical truths, but rather it was the proper role of the sciences (both pure and applied) to seek out justification in terms of the fundamental truths that may or may not appear to be obvious to our intuitions.

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Writings in English

Theory of science, attempt at a detailed and in the main novel exposition of logic with constant attention to earlier authors. (Edited and translated by Rolf George University of California Press, Berkeley and Los Angeles 1972)
Theory of science (Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell - D. Reidel Publishing Company, Dordrecht and Boston 1973)
Ewald, William B., ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford University Press, 1996.
1810. Contributions to a better grounded presentation of mathematics, 174-225.
1817. Purely analytic proof of the theorem that between any two values which give results of opposite sing, there lies at least one real root of the equation, 225-48.
1851. Paradoxes of the Infinite, 249-92 (excerpt).

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