Further reading Leitura Bolzano, the priest who took refuge in mathematics Bernard Bolzano (1781–1848) was born and died in Prague, then part of the Austrian Empire. He was a Catholic priest and professor of philosophy of religion, but his ideas — pacifist and critical of the social order — displeased the Church and the government: in 1819 he lost his professorship and began living away, writing. Mathematics became his refuge, and his contemporaries hardly read him. Bernard Bolzano.Unidentified author · Wikimedia Commons · Public domain In 1817 he published Rein analytischer Beweis (“Purely Analytical Proof”), in which he demonstrated without resorting to drawings that a continuous function that changes sign in an interval cancels out there — the current intermediate value theorem. To do this, he needed a definition of continuity that did not depend on geometric intuition. Bolzano realized the need for rigor in analysis so early that Felix Klein would call him the “father of arithmetization.” He and Cauchy never met, but they arrived at very similar definitions of limit, continuity and convergence; it was Cauchy, with more prestige, who made them known. In 1834, Bolzano constructed a function continuous on an interval and with no derivative at any point of it. The example was unpublished, and the credit ended up going to Weierstrass, who rediscovered these results decades later. Other works suffered the same fate: today we call it the Bolzano-Weierstrass theorem which guarantees that every limited set with infinite elements has at least one accumulation point, and one of the series convergence criteria is named after Cauchy. In the posthumous book Paradoxes of Infinity (1851), Bolzano showed that there are as many real numbers between 0 and 1 as between 0 and 2 — just associate $x$ with $2x$ —, or as many points in a segment of one centimeter or two. Resuming an observation by Galileo, he accepted that an infinite set can be put in correspondence with a part of itself. He seems to have intuited that the infinity of real numbers is of another kind than that of integers, an idea that only Cantor would make precise. It is not for nothing that it is said that Bolzano was “a voice crying in the desert”.