# Bernhard Bolzano

Bernhard Bolzano was born and died in Prague, Czechoslovakia. Although he was a priest, he had ideas contrary to those of the Church. His mathematical discoveries were very little recognized by his contemporaries. In 1817 published the book "Rein Analytisches Beweis" (purely analytical proof), proving by arithmetic methods the algebra location theorem, requiring for this a non-geometric concept of continuity of a curve or function.

Bolzano had by then understood the need for rigor in Analysis so well that Klein called him "the father of arithmeticization," though he had less influence than Cauchy with his analysis based on geometric concepts. Although the two had never met, their definitions of boundary, derivative, continuity, and convergence were quite similar.

In a posthumous work of 1850, Bolzano even stated important properties of finite sets and, relying on Galileo's theories, showed that there are as many real numbers between 0 and 1, as between 0 and 2, or as many in a straight line segment. one centimeter as well as a two-centimeter line segment. It seems to have realized that the infinity of real numbers is of a different type from the infinity of whole numbers, being non-enumerable, being closer to modern mathematics than any of its contemporaries.

In 1834 Bolzano had imagined a continuous function in a range that had not been derived at any point in that range, but the example given was not known in his day, and all merits were given to Wieirstrass who was busy rediscovering these results after fifty. years. We know today as Bolzano-Weierstrass theorem that a limited set containing infinite elements, points, or numbers has at least one point of accumulation. The same happened with the infinite series convergence criteria that now bear the name of Cauchy and so on with other results. Some say that Bolzano was "a voice crying out in the desert."

Source: Fundamentals of Elementary Mathematics, Gelson Iezzi - Current Publisher