Bernard Bolzano (UK: /bɒlˈtsɑːnoʊ/, US: /boʊltˈsɑː-, boʊlˈzɑː-/; German: [bɔlˈtsaːno]; Italian: [bolˈtsaːno]; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848)[5] was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal views.

When he was ten years old, Bolzano entered the Gymnasium of the Piarists in Prague, which he attended from 1791 to 1796.

[7] Bolzano entered the University of Prague in 1796 and studied mathematics, philosophy and physics.

[5] He proved to be a popular lecturer not only in religion but also in philosophy, and he was elected Dean of the Philosophical Faculty in 1818.

Bolzano alienated many faculty and church leaders with his teachings of the social waste of militarism and the needlessness of war.

His political convictions, which he was inclined to share with others with some frequency, eventually proved to be too liberal for the Austrian authorities.

On December 24, 1819, he was removed from his professorship (upon his refusal to recant his beliefs) and was exiled to the countryside and then devoted his energies to his writings on social, religious, philosophical, and mathematical matters.

[8] To this end, he was one of the earliest mathematicians to begin instilling rigor into mathematical analysis with his three chief mathematical works Beyträge zu einer begründeteren Darstellung der Mathematik (1810), Der binomische Lehrsatz (1816) and Rein analytischer Beweis (1817).

These works presented "...a sample of a new way of developing analysis", whose ultimate goal would not be realized until some fifty years later when they came to the attention of Karl Weierstrass.

[10] Like several others of his day, he was skeptical[dubious – discuss] of the possibility of Gottfried Leibniz's infinitesimals, that had been the earliest putative foundation for differential calculus.

Bolzano's notion of a limit was similar to the modern one: that a limit, rather than being a relation among infinitesimals, must instead be cast in terms of how the dependent variable approaches a definite quantity as the independent variable approaches some other definite quantity.

Bolzano also gave the first purely analytic proof of the fundamental theorem of algebra, which had originally been proven by Gauss from geometrical considerations.

Today he is mostly remembered for the Bolzano–Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano's first proof and which was initially called the Weierstrass theorem until Bolzano's earlier work was rediscovered.

[11] Bolzano's posthumously published work Paradoxien des Unendlichen (The Paradoxes of the Infinite) (1851) was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor, and Richard Dedekind.

Bolzano also did valuable work in mathematics, which remained virtually unknown until Otto Stolz rediscovered many of his lost journal articles and republished them in 1881.

In his 1837 Wissenschaftslehre Bolzano attempted to provide logical foundations for all sciences, building on abstractions like part-relation, abstract objects, attributes, sentence-shapes, ideas and propositions in themselves, sums and sets, collections, substances, adherences, subjective ideas, judgments, and sentence-occurrences.

These attempts were an extension of his earlier thoughts in the philosophy of mathematics, for example his 1810 Beiträge where he emphasized the distinction between the objective relationship between logical consequences and our subjective recognition of these connections.

Human knowledge, he states, is made of all truths (or true propositions) that men know or have known.

The written sentence does have existence (it has a certain location at a certain time, say it is on your computer screen at this very moment) and expresses the proposition in itself which is in the realm of in itself (i.e. an sich).

An intuition is a simple idea, it has only one object (Einzelvorstellung), but besides that, it is also unique (Bolzano needs this to explain sensation).

Intuitions (Anschauungen) are objective ideas, they belong to the an sich realm, which means that they don't have existence.

Bolzano explains that this change, in your mind, is essentially a simple idea (Vorstellung), like, 'this smell' (of this particular rose).

[15] According to Bolzano, all propositions are composed out of three (simple or complex) elements: a subject, a predicate and a copula.

Logically analytical propositions, for instance, are those in which all the non-logical parts can be replaced without change of truth value.

Two propositions are 'compatible' (verträglich) with respect to one of their component parts x if there is at least one term that can be inserted that would make both true.

V. Improper meaning: True signifies that some object is in reality what some denomination states it to be.

Mere presentations or thoughts are examples of mental activities which do not necessarily need to be stated (behaupten), and so are not judgments (§ 34).

[3] Alois Höfler (1853–1922), a former student of Franz Brentano and Alexius Meinong, who subsequently become professor of pedagogy at the University of Vienna, created the "missing link between the Vienna Circle and the Bolzano tradition in Austria.

"[17] Bolzano's work was rediscovered, however, by Edmund Husserl[18] and Kazimierz Twardowski,[19] both students of Brentano.

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.