Bolzano, Bernhard

Born Oct. 5,1781, in Prague; died there on Dec. 18, 1848. Czech mathematician, philosopher, and theologian.

Bolzano graduated from the faculties of philosophy (1800) and theology (1805) of Charles University (Prague). From 1805 to 1820 he occupied the chair of the history of religion at that university. He was dismissed in 1820 because he was a freethinker, and he was deprived of the right to lecture in public. After this, he worked mainly in the field of logic and mathematics.

Bolzano’s principal work, The Study of Science (1837), is an extensive historical and critical survey of traditional doctrines of logic with an original exposition of logic. Bolzano worked a great deal on the logical bases of mathematical analysis and was the first (1817) to advance the idea of the arithmetical theory of real numbers. A number of fundamental concepts and theorems of analysis that are usually associated with later studies by other mathematicians are found in his works (those published during his lifetime and those included in his legacy of manuscripts). In The Paradoxes of the Infinite (published in 1851), Bolzano anticipated G. Cantor’s study of infinite sets. Proceeding from G. Leibniz’ idea, Bolzano defended the objective existence of the actually infinite. Moreover, he distinguished between two kinds of existence of the objective: real existence—the “directly given”—and unreal but possible existence— existence “in itself.” The possibility of unreal, objective existence does not depend on subjective knowledge and is created not by thinking (according to Bolzano, the possibility of thinking a thing in no way constitutes a basis for the possibility of its existence), but by “pure concepts,” which play the role of a determining principle for all the real as well as for all the objectively possible. Insofar as the truths that follow from the pure concepts are objectively possible, the existence of infinite sets, “at least among things which are unreal,” is also objectively possible. For example, there exists an infinite set of “all truths in themselves.” A combination of dialectical automism (simple substances that are constantly changing and interacting) and Platonism (the doctrines of “truths in themselves,” “pure concepts,” and so forth), in which the ideas of ancient authors are transformed in accordance with the scientific practice of “modern times,” is characteristic of Bolzano’s philosophy as a whole.