Bernard bolzano

Bernard Placidus Johann Gonzal Nepomuk Bolzano (Prague, Bohemia (now the Czech Republic), October 5, 1781 – idem, December 18, 1848), known as Bernard Bolzano was a Bohemian mathematician, logician, philosopher and theologian who wrote in German and who made important contributions to mathematics and the theory of knowledge.

In mathematics, he is known for Bolzano's theorem, as well as the Bolzano-Weierstrass theorem, which he outlined as a lemma for another work in 1817, and was developed decades later by Karl Weierstrass

In his philosophy, Bolzano criticized the idealism of Hegel and Kant, stating that numbers, ideas, and truths exist independently of the people who think them.

Biography

In 1796 Bolzano enrolled in the Faculty of Philosophy at the University of Prague. During his studies he wrote: "My special predilection for Mathematics is based in a particular way on its speculative aspects, in other words, I appreciate very much the part of Mathematics that is at the same time Philosophy. &#3. 4; In the autumn of 1800 he began to study Theology. He devoted himself to it for the next three years, during which he also prepared his doctoral thesis in Geometry. He obtained his doctorate in 1804, after having written a thesis in which he expressed his opinion on Mathematics and on the characteristics of a correct mathematical proof. In the foreword he wrote: & # 34; I could not be satisfied by a strictly rigorous proof, if it did not derive from the concepts contained in the thesis to be proved. & # 34;

Two years after being made a doctor, Bolzano was ordained as a Catholic priest. However, his true vocation was teaching, and in 1804 he obtained the chair of Philosophy and Religion at the University of Prague. In relation to this chair, it should be noted that at that time, due to the expansion of enthusiasm aroused by the French Revolution, the first political movements that claimed freedom of thought and the independence of national communities had developed. These demands were of great concern to the authoritarian states, and especially to the Austrian Empire, within whose limits numerous very different ethnic groups were integrated, among which nationalist movements were being born. To counteract these movements, the Austrian Empire, in agreement with the Catholic Church, which was clearly aligned with conservative positions compared to those coming from the French Revolution, carried out a series of initiatives. Among these was to establish a chair of Philosophy of Religion in each University, which would be erected as a bulwark against freedom of thought and against nationalist positions.

However, the appointment of Bolzano to occupy said chair at the University of Prague did not have the success that the authorities expected. His teachings were permeated by strong pacifist ideals and by a lively demand for political justice. In addition, Bolzano enjoyed, due to his intellectual qualities, enormous prestige among his fellow teachers and among students. After some pressure from the Austrian government, in 1819 Bolzano was dismissed from his professorship. Due to his personality, he did not accept this dismissal without expressing his disagreement, with which he was suspended, on a charge 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 occupy an important role in the intellectual life of his country.

Bolzano wrote in 1810 Beiträge zu einer begründeteren Darstellung der Mathematik. Erste Lieferung, the first in a scheduled series of writings on the foundations of mathematics. In the second part we find Der binomische Lehrsatzl... from 1816 and Rein analytischer Beweis... (Pure Mathematical Demonstration) from 1817, which contain an attempt to set up calculus. infinitesimal that does not use the concept of infinitesimal. In the prologue to the first of both, he declares that his work is an example of the new way of developing analysis. Despite the fact that Bolzano managed to prove exactly everything he declared, his theories were only understood after his death. In the work of 1817 Bolzano understood that he freed the concepts of limit, convergence and derivative from geometric notions, replacing them with purely arithmetic and numerical concepts. Bolzano was aware of the existence of a deeper problem: it was necessary to refine and enrich the very concept of number. In this work it is necessary to situate the proof of the intermediate value theorem with the new approximation of Bolzano, and the one that was also called the Cauchy series. This concept appears in a work by Cauchy, published four years later, although it is unlikely that the French mathematician was aware of Bolzano's work.

After 1817, Bolzano spent many years without publishing anything related to mathematics. However, in 1837, he published Wissenschaftslehre , an attempt to elaborate a complete theory of knowledge and science. Bolzano tried to provide logical foundations to all sciences, built starting from abstractions, abstract objects, attributes, constructions of demonstrations, links... Most of these attempts take up those previous works that affect the objective relationship between the consequences logical (things as they occur) and our purely subjective perception of these consequences (our way of approaching the facts). Here he approaches the philosophy of mathematics. One of the basic notions in Bolzano's Wissenschaftslehre is the so-called „Satz an sich.” He first introduces the notions of proposition and representation. "The grass is green" is a proposition (Satz): in this connection of words, something is said or affirmed. "Grass," however, is only a representation (Vorstellung). Something is represented by it, although it does not affirm anything. Bolzano's notion of a proposition is quite broad: "A rectangle is round" it is a proposition, even if it is false by virtue of self-contradiction, because it is intelligibly composed out of intelligible parts. Bolzano does not present a complete definition of Satz an sich (that is, of a proposition itself) but he does give us enough information to understand what he means by it. A proposition itself has no existence, it is true or false, regardless of whether someone knows or thinks it is true or false, and that is what they "get" thinking beings. So a sentence written 'Socrates is wise' captures a proposition itself, namely the content 'Socrates is wise'. The written sentence has existence since he is reading it right now at this very moment and expresses the proposition itself that is in the realm of itself. From this distinction a gap is generated with the Kantian philosophy in the Austrian tradition. For Bolzano, we have no certainty regarding the truths, or those supposed as such, of nature or of mathematics, and precisely the role of the sciences, both pure and applied, is to find a justification of the truths (or of the fundamental laws) that often contradict our intuitions. Many scholars, including Edmund Husserl, consider this text the first major work on logic and problems of knowledge after Leibnitz's.

Between 1830 and 1840, Bolzano worked on a major work, Grössenlehre, in which he tried to reinterpret all of mathematics under logical grounds. He only published part of it, hoping that his students would continue his work and publish a full version. Bolzano died in Prague on December 18, 1848. In 1854, three years after his death, a student of his published Bolzano's work Paradoxien des Unendlichen , a study on the paradoxes of infinity. The term "set" appears for the first time, in the German form Menge. In this work Bolzano provides examples of one-to-one correspondence between the elements of an infinite set and a proper subset of it, which allows characterizing the concept of infinite set.

Most of Bolzano's work remained in manuscript form, so it had a very small circulation and little influence on the development of the subject. Many of his works were not published until 1862 and even later. Bolzano's theories on mathematical infinity anticipated Georg Cantor's on infinite sets.