Bernhard Bolzano

Bernhard 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. In autumn of 1800 he began 3 years of theological study. While pursuing these 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.  During the same year he joined the theology department at the university and was ordained a Catholic priest.  His dedication to the Church did not hinder his mathematical interests or cause him to give up. This must have been a crutial year for Bolzano’s career to develop because he was also appointed to the chair of philosophy and religion 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.  Because of his pacifist beliefs and his concern for economic justice, Bolzano was suspended from his position in 1819 after pressure from the Austrian government.  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 didn’t give up without a fight. 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. Austrian government had an extremely tight grasp on what Bolzano could and could not do. After leaving the university, he moved to the small village of Techobuz , where he stayed until 1842. 

Though it was risky, he continued to write and to play an important role in the intellectual life of his country.  During this time Bolzano wrote the first of an intended series on the foundations of mathematics. Bolzano wrote the second of his series but did not publish it. Instead he decided to

“... make myself better known to the learned world by publishing some papers which, by their titles, would be more suited to arouse attention.”

Pursuing this idea, he published two papers, which contain an attempt to free calculus from the concept of the infinitesimal.  He was successful.  Infinitesimal is known as the opposite of infinity. If we use fractions it is quite small!

The papers are, as Bolzano stated, “ a sample of a new way of developing analysis.”

One of the two papers published was pure analytical proof (1817).  This paper gives a proof of the intermediate value theorem with Bolanzo’s new approach.  These papers define what is now called the Cauchy sequence.  The idea appears in Cauchy's work four years later but it is highly unlikely that Cauchy had read anything Bolzano published.

After publishing the paper in 1817, Bolzano didn’t publish any mathematical works until 1837 when he published Wissenschaftslehre.  This was an attempt at a complete theory of science and knowledge.

Between 1830 and the 1840s, Bolzano worked on a major piece of his career Grössenlehre. This was an attempt to put the whole of mathematics on a logical foundation. Grössenlehre was published in several parts. Bolzano never quite finished but he hoped that his students would finish and publish the complete work.

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’s work covered three main 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 complete them because he lacked the proper mathematical tool of topology, which would not be invented until a later date.  Bolzano did establish definitions for basic geometric concepts and was the first person to state the Jordan curve theorem.   Which states 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.  This concept stumped several previous mathematicians. Although he did not succeed, he did come up with some important discoveries including the Bolzano-Weierstrass theorem. In addition, he recognized some of the paradoxical qualities of infinite sets, a breakthrough which he did not investigate and would later be 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.