Bernard Bolzano

Family

Bolzano was the son of two pious Catholics. His father, Bernard Pompeius Bolzano, was an Italian who had moved to Prague, where he married Maria Cecilia Maurer who came from Prague's German-speaking family Maurer. Only two of their twelve children lived to adulthood.

Career

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

Bolzano entered the University of Prague in 1796 and studied mathematics, philosophy and physics. Starting in 1800, he also began studying theology, becoming a Catholic priest in 1804. He was appointed to the new chair of philosophy of religion at Prague University in 1805. 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. He urged a total reform of the educational, social and economic systems that would direct the nation's interests toward peace rather than toward armed conflict between nations. 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 24 December 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.

Although forbidden to publish in mainstream journals as a condition of his exile, Bolzano continued to develop his ideas and publish them either on his own or in obscure Eastern European journals. In 1842 he moved back to Prague, where he died in 1848.

Mathematical work

Bolzano made several original contributions to mathematics. His overall philosophical stance was that, contrary to much of the prevailing mathematics of the era, it was better not to introduce intuitive ideas such as time and motion into mathematics. 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.

To the foundations of mathematical analysis he contributed the introduction of a fully rigorous ε–δ definition of a mathematical limit. Bolzano was the first to recognize the greatest lower bound property of the real numbers. Like several others of his day, he was skeptical 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. He also gave the first purely analytic proof of the intermediate value theorem (also known as Bolzano's theorem). 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.

Philosophical work

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's main claim to fame, however, is his 1837 Wissenschaftslehre (Theory of Science), a work in four volumes that covered not only philosophy of science in the modern sense but also logic, epistemology and scientific pedagogy. The logical theory that Bolzano developed in this work has come to be acknowledged as ground-breaking. Other works are a four-volume Lehrbuch der Religionswissenschaft (Textbook of the Science of Religion) and the metaphysical work Athanasia, a defense of the immortality of the soul. 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.

Wissenschaftslehre (Theory of Science)

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. For Bolzano, it was not enough that we merely have confirmation of natural or mathematical truths, but rather it was the proper role of the sciences (both pure and applied) to seek out justification in terms of the fundamental truths that may or may not appear to be obvious to our intuitions.

Bolzano begins his work by explaining what he means by theory of science, and the relation between our knowledge, truths and sciences. Human knowledge, he states, is made of all truths (or true propositions) that men know or have known. However, this is a very small fraction of all the truths that exist, although still too much for one human being to comprehend. Therefore, our knowledge is divided into more accessible parts. Such a collection of truths is what Bolzano calls a science (Wissenschaft). Not all true propositions of a science have to be known to men; hence, this is how we can make discoveries in a science.

To better understand and comprehend the truths of a science, men have created textbooks (Lehrbuch), which of course contain only the true propositions of the science known to men. But how to know where to divide our knowledge, that is, which truths belong together? Bolzano explains that we will ultimately know this through some reflection, but that the resulting rules of how to divide our knowledge into sciences will be a science in itself. This science, that tells us which truths belong together and should be explained in a textbook, is the Theory of Science (Wissenschaftslehre).

In the Wissenschaftslehre, Bolzano is mainly concerned with three realms:

(1) The realm of language, consisting in words and sentences.

Philosophical legacy

Bolzano came to be surrounded by a circle of friends and pupils who spread his thoughts about (the so-called Bolzano Circle), but the effect of his thought on philosophy initially seemed destined to be slight.

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." Bolzano's work was rediscovered, however, by Edmund Husserl and Kazimierz Twardowski, both students of Brentano. Through them, Bolzano became a formative influence on both phenomenology and analytic philosophy.

Writings

Bolzano: Gesamtausgabe (Bolzano: Collected Works), critical edition edited by Eduard Winter, Jan Berg, Friedrich Kambartel, Bob van Rootselaar, Stuttgart: Fromman-Holzboog, 1969ff. (103 Volumes available, 28 Volumes in preparation).

Wissenschaftslehre, 4 vols., 2nd rev. ed. by W. Schultz, Leipzig I–II 1929, III 1980, IV 1931; Critical Edition edited by Jan Berg: Bolzano's Gesamtausgabe, vols. 11–14 (1985–2000).

Bernard Bolzano's Grundlegung der Logik. Ausgewählte Paragraphen aus der Wissenschaftslehre, Vols. 1 and 2, with supplementary text summaries, an introduction and indices, edited by F. Kambartel, Hamburg, 1963, 1978².

Bolzano, Bernard (1810), Beyträge zu einer begründeteren Darstellung der Mathematik. Erste Lieferung (Contributions to a better grounded presentation of mathematics; Ewald 1996, pp. 174–224 and The Mathematical Works of Bernard Bolzano, 2004, pp. 83–137).

Bolzano, Bernard (1817), Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege, Wilhelm Engelmann (Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation; Ewald 1996, pp. 225–48.

Franz Prihonsky (1850), Der Neue Anti-Kant, Bautzen (an assessment of the Critique of Pure Reason by Bolzano, published posthumously by his friend F. Prihonsky).*Bolzano, Bernard (1851), Paradoxien des Unendlichen, C.H. Reclam (Paradoxes of the Infinite; Ewald 1996, pp. 249–92 (excerpt)).

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.

Translations and compilations

Theory of Science (selection edited and translated by Rolf George, Berkeley and Los Angeles: University of California Press, 1972).

Theory of Science (selection edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell, Dordrecht and Boston: D. Reidel Publishing Company, 1973).

Theory of Science, first complete English translation in four volumes by Rolf George and Paul Rusnock, New York: Oxford University Press, 2014.

The Mathematical Works of Bernard Bolzano, translated and edited by Steve Russ, New York: Oxford University Press, 2004 (re-printed 2006).

On the Mathematical Method and Correspondence with Exner, translated by Rolf George and Paul Rusnock, Amsterdam: Rodopi, 2004.

Selected Writings on Ethics and Politics, translated by Rolf George and Paul Rusnock, Amsterdam: Rodopi, 2007.

Franz Prihonsky, The New Anti-Kant, edited by Sandra Lapointe and Clinton Tolley, New York, Palgrave Macmillan, 2014.

Russ, S. B. (1980). "A translation of Bolzano's paper on the intermediate value theorem". Historia Mathematica. 7 (2): 156–185. doi:10.1016/0315-0860(80)90036-1. (Translation of Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (Prague 1817))