Bernard Placidus Johann Nepomuk Bolzano. Known as Bernard Bolzano , he was a Czech mathematician, philosopher and theologian who made significant contributions to both mathematics and the Theory of Science , in some respects constituting an interesting precedent for mathematical logic. He freed calculus from the infinitesimal concept and gave examples of the correspondence of functions 1-1.
Summary
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- 1 Biographical data
- 1 Death
- 2 Contributions to mathematics
- 3 Publications
- 4 Sources
Biographical data
He was born in Prague on October 5 , 1781 . Son of Bernard Pompeius Bolzano a German antiquarian and Maria Cecilia Maurer of Italian origin.
Despite the large family, Bernard and one of his brothers, Johann, were the only two to reach adulthood. His health, however, was delicate and he had to fight respiratory problems throughout his life.
Bolzano’s upbringing was an important factor in the ideas he later taught, he was greatly influenced by his father’s active attempts to care for his fellow man.
He entered the faculty of philosophy at the University of Prague in 1796 , studied philosophy, mathematics and physics . Two years after being made a doctor, he was ordained a Roman Catholic priest in 1805 . Professor of religion in Prague and amateur mathematician. In 1820 the authorities prohibited him from exercising any academic activity because of his critical position with respect to the social conditions in force in the Austro-Hungarian Empire.
Bolzano’s appointment to that chair at the University of Prague was not as successful as the authorities hoped. His teachings were permeated by strong pacifist ideals and a lively demand for political justice. In addition, Bolzano enjoyed, due to his intellectual qualities, enormous prestige among his fellow professors and among students.
After some pressure from the Austrian government, in 1819 he was removed from his chair. Due to his personality, he did not accept this dismissal without expressing his disagreement, with which he was suspended, under an accusation of heresy, placed under house arrest and prohibited from publishing. Despite government censorship, his books were published outside the Austrian Empire and Bolzano continued to write and play an important role in the intellectual life of his country.
He was particularly influenced in his mathematical studies by reading Kaestner ‘s Mathematische Anfangsgründe . Erich Kaestner was concerned with the philosophical questions of mathematics, and was deeply interested in the philosophy of mathematics, he was very careful to demonstrate many results that are believed to be “evident”, so that they do not require proof, by other mathematicians of the epoch. Bolzano, who soon developed a strong belief in this approach, wrote:
It has been a special pleasure in mathematics rested therefore, especially in its purely speculative parts, that is to say that only the appreciated part of mathematics that was in philosophy at the same time.
His works took many decades to be known. Most of Bolzano’s works remained in manuscript and therefore did not influence the development of the subject. Many of his works were not published until 1862 or later. It is also notable that you have given an example of a function that is differentiable nowhere yet continues everywhere.
Although some of his books had to be published outside of the Austrian Empire, due to government censorship, he continued to write and play an important role in the intellectual life of his country. In fact, he had won a partial lifting of the publication ban, and was prohibited from publishing only something of a religious or political nature.
Death
He died in Prague on December 18 , 1848 at the age of 67.
Contributions to mathematics
Bolzano’s scientific concerns were far advanced for his time, concerned as he was with the foundations of various branches of mathematics. In this science the following contributions can be highlighted:
- Bolzano’s theorem
- Bolzano-Weierstrass theorem
- He began the process of placing the analysis on a more rigorous basis. Precursor to the arithmeticization of analysis.
- He was the first to find a continuous function at all points of an interval but not differentiable at any of them.
- The criterion of convergence of sequences and infinite series attributed to Cauchy is due to him.
- He devoted himself to the study of the paradoxes of infinity
- Established one-to-one correspondence between an infinite set and its own subset
- Fixed the concept of distance
- He was one of the forerunners of set theory and modern logic
- He was one of the first to separate logic from psychology
- He was the first to give a precise definition of the idea and concept of limit as a support to define the derivative and the integral.
Bolzano’s scientific concerns were far advanced for his time, concerned as he was with the foundations of various branches of mathematics, namely the theory of functions, logic, and the notion of the cardinal. After proving the intermediate value theorem, he gave the first example of a non-differentiable continuous function on the set of real numbers. In the field of logic, he dealt with the truth table of a proposition and introduced the first working definition of deductibility. He also studied, before Georg Cantor , infinite sets.
Publications
- Contributions to a more solid presentation of mathematics ( 1810 ).
- Purely analytical proof of the theorem that says that between two values that give results of opposite sign, there is at least one real solution of equation ( 1817 )
- “Theory of functions” ( 1834 ) with a purely arithmetic proof (until then only the geometric one was known) of the mean value theorem.
- “Theory of science” ( 1837 ). 4 volumes
- “Essay for a new presentation of logic” (1837)
- He has a posthumous work ” Paradoxes of infinity” ( 1851 ), which was published by one of his students.
- “Der binomische Lehrsatz” ( 1816 )
- “Pure Analytical Test” (1817)
