Bernard Bolzano

Bernard (Bernhard) Placidus Johann Nepomuk Bolzano ((1781-10-05)October 5, 1781 – December 18, 1848) was a Bohemian mathematician, theologian, philosopher, logician and antimilitarist of German mother tongue.

Family

Bolzano was the son of two pious Catholics. His father, Bernard Pompeius Bolzano, was born in northern Italy and moved to Prague, where he married Maria Cecelia Maurer, the (German-speaking) daughter of a Prague merchant. Only two of their twelve children lived to adulthood.

Career

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 then newly created chair of philosophy of religion in 1805. He proved to be a popular lecturer not just in religion but also philosophy, and was elected head of the philosophy department 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. Upon his refusal to recant his beliefs, Bolzano was dismissed from the university in 1819. His political convictions (which he was inclined to share with others with some frequency) eventually proved to be too liberal for the Austrian authorities. He exiled to the countryside and at that point 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.

Works

Bolzano's early work Paradoxien des Unendlichen (The Paradoxes of the Infinite) was greatly admired by many of the eminent logicians who came after him, including Charles 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 study 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 basically 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.

Metaphysics

Bolzano's metaphysical system, as he describes it in the Wissenschaftslehre, is composed of four realms:

(1) The realm of language, consisting in words and sentences. (2) The realm of thought, consisting in subjective ideas and judgments. (3) The realm of logic, consisting in objective ideas and propositions in themselves. (4) The realm of all objects, which also contains the other three realms and divides into attributes and pure objects.

Bolzano devotes a great part of the Wissenschaftslehre to an explanation of these four realms and their relations. Two distinctions play a prominent role in his system. Firstly, each realm divides into parts and wholes. Words are parts of sentences, subjective ideas are parts of judgments, objective ideas are parts of propositions in themselves, and attributes are parts of pure objects. Secondly, all objects divide into those that exist, and those that are in themselves. Bolzano's original claim is that the logical realm is populated by objects of the latter kind.

Satz an Sich (Proposition in itself)

Satz an Sich is a basic notion in Bolzano's Wissenschaftslehre. It is introduced at the very beginning, in section 19. Before giving a definition, Bolzano first introduces the notions of proposition (spoken or written or otherwise) and idea. "The grass is green" is a proposition (Satz): in this connection of words, something is said or asserted. "Green grass", however, is only an idea (Vorstellung). Something is represented by it, but it does not say or assert anything. Bolzano's notion of proposition is fairly broad: "A rectangle is round" counts a proposition, even though it is false by virtue of self-contradiction, because it is composed in an intelligible manner out of intelligible parts. A Satz an Sich is what is thought when one thinks about a proposition and can still ask oneself whether or not this proposition has been said or thought by someone or not. Hence a Satz an Sich states that something is or isn't, with no condition on it being true or not or on it being spoken, thought etc. or not. Bolzano's use of the term an sich differs greatly from that of Kant; for his use of the term see an sich.

Logic

According to Bolzano, all propositions are composed out of three (simple or complex) elements: a subject, a predicate and a copula. Instead of the more traditional copulative term 'is', Bolzano prefers 'has'. The reason for this is that 'has', unlike 'is', can connect a concrete term, such as 'Socrates', to an abstract term such as 'baldness'. "Socrates has baldness" is, according to Bolzano, preferable to "Socrates is bald" because the latter form is less basic: 'bald' is itself composed of the elements 'something', 'that', 'has' and 'baldness'. Bolzano also reduces existential propositions to this form: "Socrates exists" would simply become "Socrates has existence (Dasein)".

A starring role in Bolzano’s logical theory is played by the notion of variations: various logical relations are defined in terms of the changes in truth value that propositions incur when their non-logical parts are replaced by others. 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' (vertraglich) 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. A proposition Q is 'deducible' (ableitbar) from a proposition P, with respect to certain of their non-logical parts, if any replacement of those parts that makes P true also makes Q true. If a proposition is deducible from another with respect to all its non-logical parts, it is said to be 'logically deducible'. Besides the relation of deducibility, Bolzano also has a stricter relation of 'consequentiality' (Abfolge). This is an asymmetric relation that obtains between true propositions, when one of the propositions is not only deducible from, but also explained by the other.

Mathematics

Bolzano made several original contributions to mathematics. In Parallelogram area theory he demonstrated that for similar rhombi, the ratio of the area of rhombus A to the area of rhombus B is equal to the square of the ratio of the width of A to the width of B. To the foundations of mathematical analysis he contributed the introduction of a fully rigorous ε-δ definition of a mathematical limit and 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 legacy

Due to the fact that Bolzano's most important work, the Wissenschaftslehre, could not be published during his lifetime, the impact of his thought on philosophy initially seemed destined to be slight. His work was rediscovered, however, by Edmund Husserl and Kazimierz Twardowski, both students of Franz Brentano. Through them, and through Gottlob Frege, also an admirer, Bolzano became a formative influence on both phenomenology and analytic philosophy.

Writings in English

Theory of science, attempt at a detailed and in the main novel exposition of logic with constant attention to earlier authors. (Edited and translated by Rolf George University of California Press, Berkeley and Los Angeles 1972)
Theory of science (Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell - D. Reidel Publishing Company, Dordrecht and Boston 1973)
Ewald, William B., ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford University Press, 1996 contains the three following essays:
1810. Contributions to a better grounded presentation of mathematics, 174-224.
1817. 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, 225-48.
1851. Paradoxes of the Infinite, 249-92 (excerpt).
Paradoxes of the infinite - Translated from the German of the posthumous edition by Fr. Prihonský and furnished with a historical introduction by Donald A. Steele - Routledge & Kegan Paul, 1950.
On the mathematical method and correspondence with Exner - Translated by Paul Rusnock and Rolf George - Amsterdam, Rodopi, 2004.
The mathematical works of Bernard Bolzano - Edited by Steve Russ - Oxford, Oxford University Press, 2004.
Selected Writings on Ethics and Politics - Translated by Paul Rusnock and Rolf George - Amsterdam, Rodopi, 2007.

External links

O'Connor, John J.; Robertson, Edmund F., "Bernard Bolzano", MacTutor History of Mathematics Archive, University of St Andrews
Biography of Bernard Bolzano
Bernard Bolzano's Theory of Science

References

Künne, Wolfgang. (1998). "Bolzano, Bernard". Routledge Encyclopedia of Philosophy 1: 823-827. London: Routledge. Retrieved on 2007-03-05