9. Historical Tidbits

9.4. Bolzano, Bernhard (1781-1848)

Bernard Bolzano was a philosopher and mathematician whose contributions were not fully recognized until long after his death. He is especially important in the fields of logic, geometry and the theory of real numbers.

Bernardus Placidus Johann Nepomuk Bolzano was born in Prague, Bohemia (now part of the Czech Republic), on October 5, 1781. His father was an art dealer and both parents were very pious Christians. Coming from such a religious household, Bernard grew up with a high moral code and a belief in holding to his principles. It was this background that attracted him to the Church and the priestly life.

Bolzano entered the University of Prague in 1796, where he studied philosophy, mathematics and physics. After graduation, he joined the theology department at the university and was ordained a Catholic priest in 1804. Despite his dedication to the Church, he did not give up his mathematical interests and was at one time recommended for the chair of the mathematics department.

The year 1805 started a struggle that would dominate the rest of his life. In a political move, the Austrian-Hungarian Empire set up a chair in the philosophy of religion at each university. The empire was comprised of many different ethic groups that were prone to nationalistic movements for independence. Spurred by the "free thinking" of the recent French Revolution, these movements were becoming a serious problem to holding the empire together. The creation of the chair was part of a greater plan to support the Catholic Church. The authorities considered the Church to be conservative and hoped it would control the liberal thinking of the time. Bolzano was appointed to the position at the University of Prague. As far as the authorities were concerned, this was a bad idea. Bolzano, though a priest, was a "free thinker" himself and was not afraid to express his beliefs in Czech nationalism.

For the next 14 years, Bolzano taught at the university, lecturing mainly on ethics, social questions and the links between mathematics and philosophy. He was very popular with both the student body, who appreciated his straightforward expression of his beliefs, and his fellow professors, who recognized his intelligence. In 1818, he became Dean of the philosophy department. However, the Austro-Hungarian authorities became displeased with his liberal views. In 1819, he was suspended from his professorship, forbidden to publish and put under police surveillance. Bolzano refused to back down. However, despite the backing of the Church, he was unable to get his job back. In 1824, after refusing to sign an official "recantation" of his nationalistic views, he resigned his seat.

After leaving the university, he moved to the small village of Techobuz , where he stayed until 1842. He then returned to Prague to continue his philosophical and mathematical studies. He died on December 18, 1848. Bolzano had many new mathematical and logical ideas during his lifetime; however, because he was prohibited from publishing by the government, most of his writings existed only in manuscript. They were not published until 1962.

Being a philosopher, Bolzano attacked his mathematics philosophically. He believed that first clear concepts could only be obtained by using logic on basic principles and definitions. By finding the foundations, the user was guaranteed rigorous proof. Sometimes, this system gave him discoveries that were amazing. At other times, especially in mathematics, it gave him wrong answers.

Bolzano did contribute much to mathematics. His work attacked mainly three subjects: geometry, the theory of real numbers and logic. In geometry, he attempted to handle the problem of Euclid's parallel postulate. He found several problems in Euclid's reasoning but was unable to solve them because he lacked the proper mathematical tool of topology which had not yet been invented. He did establish definitions for basic geometric concepts and was the first person to state the Jordan curve theorem, that a simple closed curve divides a plane into two parts. In the theory of real numbers, he tried to find its foundation and reconcile infinite quantities, a concept that had stumped previous mathematicians. Although he did not succeed, he did come up with some important discoveries including the Bolzano-Weierstrass theorem, a modern definition of a continuous function and the non- differentiable Bolzano function. In addition, he recognized some of the paradoxical qualities of infinite sets, a breakthrough which he did not pursue and would be later stated by Cantor. In logic, his ideas were generally ignored until the modern day. Not just trying to place mathematics on a logical foundation, he went a step further and tried to place all the sciences and human thinking under its scope. In his works, he tackles basic ideas like abstract truth, human judgment and rules of science. Today, he is now considered one of the precursors to modern logic.

For related information on Bolzano, see: Georg Cantor , Euclid , Karl Weierstrass , Bolzano theorem , Bolzano-Weierstrass theorem.

Historical information compiled by Paul Golba