Bernhard Bolzano entered the Philosophy Faculty of the University of Prague in 1796, studying philosophy and mathematics. In 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. Two days after receiving his doctorate Bolzano was ordained a Roman Catholic priest. But he soon realized that teaching and not ministering defined his true vocation.

He 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. 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.

In 1810, Bolzano wrote the first of an intended series of papers on the foundations of mathematics. Bolzano wrote the second of his series but did not publish it.

He did however write a pair of papers which attempted to free calculus from the concept of the infinitesimal. 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. It is in a 1817 paper that Bolzano gives a proof of the intermediate value theorem with his new approach, and in this work he defined what is now called a Cauchy sequence.

Bolzano published hardly anything for 20 years. However, in 1837, he published an attempt at a complete theory of science and knowledge. His study of paradoxes of the infinite was published in 1851, 3 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. 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.

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.