Bernard Bolzano

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

Contents 1 Family 2 Career 3 Works 4 Wissenschaftslehre (Theory of science) 4.1 Metaphysics 4.2 Satz an Sich (Proposition in itself) 4.3 Ideas and objects 4.4 Sensation and Simple Ideas 4.5 Logic 4.6 Truth 5 Mathematics 6 Philosophical legacy 7 Writings in English 8 External links 9 References

[edit] 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.

[edit] 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.

[edit] Works

Bolzano's posthumously published work Paradoxien des Unendlichen (The Paradoxes of the Infinite) 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 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.

[edit] 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.

[edit] Metaphysics

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

(1) The realm of language, consisting in words and sentences. (2) The realm of thought, consisting in subjective ideas and judgements. (3) The realm of logic, consisting in objective ideas (or ideas in themselves) and propositions in themselves.

Bolzano devotes a great part of the Wissenschaftslehre to an explanation of these realms and their relations.

Two distinctions play a prominent role in his system. Firstly, the distinction between parts and wholes. For instance, words are parts of sentences, subjective ideas are parts of judgments, objective ideas are parts of propositions in themselves. Secondly, all objects divide into those that exist, which means that they are causally connected and located and time and/or space, and those that do not exist. Bolzano's original claim is that the logical realm is populated by objects of the latter kind.

[edit] 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. Bolzano first introduces the notions of proposition (spoken or written or thought or in itself) and idea(spoken or written or thought or in itself). "The grass is green" is a proposition (Satz): in this connection of words, something is said or asserted. "grass", however, is only an idea (Vorstellung). Something is represented by it, but it does not assert anything. Bolzano's notion of proposition is fairly broad: "A rectangle is round" is a proposition - even though it is false by virtue of self-contradiction - because it is composed in an intelligible manner out of intelligible parts.

Bolzano does not give a complete definition of a Satz an Sich (i.e. proposition in itself) but he gives us just enough information to understand what he means by it. A proposition in itself (i) has no existence (that is: it has no position in time or place), (ii) is either true or false, independent of anyone knowing or thinking that it is true or false, and (iii) is what is 'grasped' by thinking beings. So a written sentence ('Socrates has wisdom') grasps a proposition in itself, namely the proposition [Socrates has wisdom]. 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). (Bolzano's use of the term an sich differs greatly from that of Kant; for Kant's use of the term see an sich.)(Bolzano, on the mathematical method, § 2)

Every proposition in itself is composed out of ideas in themselves(for simplicity, we will use 'proposition' to mean 'proposition in itself' and 'idea' to mean objective idea or idea in itself. Ideas are negatively defined as those parts of a proposition that are themselves not propositions. A proposition consists of at least three ideas, namely: a subject idea, a predicate idea and the copula (i.e. 'has', or another form of 'to have'). (Though there are propositions which contain propositions, but we won't take them into consideration right now.) Bolzano identifies certain types of ideas. There are simple ideas that have no parts (as an example Bolzano uses [something]), but there are also complex ideas that consist of other ideas (Bolzano uses the example of [nothing], which consists of the ideas [not] and [something]). Complex ideas can have the same content (i.e. the same parts) without being the same - because their components are differently connected. The idea [A black pen with blue ink] is different from the idea [A blue pen with black ink] though the parts of both ideas are the same. (Bolzano, on the mathematical method, §3)

[edit] Ideas and objects

It is important to understand that an idea does not need to have an object. Bolzano uses 'object' to denote something that is represented by an idea. An idea that has an object, represents that object. But an idea that does not have an object represents nothing. (Don't get confused here by terminology: an objectless idea is an idea without a representation.)

Let's consider, for further explanation, an example used by Bolzano. The idea [a round square], does not have an object, because the object that ought to be represented is self-contrary. A different example is the idea [nothing] which certainly does not have an object. However, the proposition [the idea of a round square has complexity] has as its subject-idea [the idea of a round square]. This subject-idea does have an object, namely a round square. But, the idea [round square] does not have an object.

Besides objectless ideas, there are ideas that have only one object, e.g. the idea [the first man on the moon] represents only one object. Bolzano calls these ideas 'singular ideas'. Obviously there are also ideas that have many objects (e.g. [the citizens of Amsterdam] and even infinitely many objects (e.g. [a prime number])(Bolzano, on the mathematical method, §4).

[edit] Sensation and Simple Ideas

Bolzano has a complex theory of how we are able to sense things. He explains sensation by means of the term intuition, in German called Anschauung. 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. As said, Bolzano’s argumentation for intuitions is by an explanation of sensation.

What happens when you sense a real existing object, for instance a rose, is this: the different aspects of the rose, like its scent and its color, cause in you a change. That change means that before and after sensing the rose, your mind is in a different state. So sensation is in fact a change in your mental state. How is this related to objects and ideas? Bolzano explains that this change, in your mind, is essentially a simple idea (Vorstellung), like, ‘this smell’ (of this particular rose). This idea represents; it has as its object the change. Besides being simple, this change must also be unique. This is because literally you can’t have the same experience twice, nor can two people, who smell the same rose at the same time, have exactly the same experience of that smell (although they will be quite alike). So each single sensation causes a single (new) unique and simple idea with a particular change as its object. Now, this idea in your mind is a subjective idea, meaning that it is in you at a particular time. It has existence. But this subjective idea must correspond to, or has as a content, an objective idea. This is where Bolzano brings in intuitions (Anschauungen); they are the simple, unique and objective ideas that correspond to our subjective ideas of changes caused by sensation. So for each single possible sensation, there is a corresponding objective idea. Schematic the whole process is like this: whenever you smell a rose, the its scent causes a change in you. This change is the object of your subjective idea of that particular smell. That subjective idea corresponds to the intuition or Anschauung. (Bolzano, Wissenschaftslehre §72).

[edit] 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.

[edit] Truth

Bolzano distinguishes five meanings the words ‘true’ and ‘truth’ have in common usage, all of which Bolzano takes to be unproblematic. The meanings are listed in order of properness.

1) Abstract objective meaning: ‘Truth’ signifies an attribute that may apply to a proposition, primarily to a proposition in itself, namely the attribute on the basis of which the proposition expresses something that in reality is as is expressed.

2) Concrete objective meaning: ‘(a) Truth’ signifies a proposition that has the attribute ‘truth’ in the abstract objective meaning. Antonym: '(a) falsity'.

3) Subjective meaning: ‘(a) Truth’ signifies a correct judgment. Antonym: '(a) mistake'.

4) Collective meaning: ‘Truth’ signifies a body or multiplicity true propositions or judgments (e.g. the biblical truth).

5) Improper meaning: ‘True’ signifies that some object is in reality what some denomination states it to be. (e.g. the true God). Antonyms: 'false, unreal, illusory'.

Bolzano’s primary concern is with the concrete objective meaning: with concrete objective truths or truths in themselves. All truths in themselves are a kind of propositions in themselves. They do not exist , i.e. they are not spatiotemporally located as thought and spoken propositions are. However, certain propositions have the attribute of being a truth in itself. Being a thought proposition is not a part of the concept of a truth in itself, notwithstanding the fact that, given God’s omniscience, all truths in themselves are also thought truths. The concepts ‘truth in itself’ and ‘thought truth’ are interchangeable, as they apply to the same objects, but they are not identical.

Bolzano offers as the correct definition of (abstract objective) truth: a proposition is true if it expresses something that applies to its object. The correct definition of a (concrete objective) truth must thus be: a truth is a proposition that expresses something that applies to its object. This definition applies to truths in themselves, rather than to thought or known truths, as none of the concepts figuring in this definition are subordinate to a concept of something mental or known.

[edit] Mathematics

Bolzano made several original contributions to mathematics. 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.

[edit] 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, Bolzano became a formative influence on both phenomenology and analytic philosophy.

[edit] 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.

[edit] External links

Wikimedia Commons has media related to: Bernard Bolzano

• O'Connor, John J.; Robertson, Edmund F., "Bernard Bolzano", MacTutor History of Mathematics archive .
• Biography of Bernard Bolzano
• Bernard Bolzano's Contributions to Logic and Ontology
• Bernard Bolzano entry in the Stanford Encyclopedia of Philosophy by Edgar Morscher
• Bolzano's Logic entry in the Stanford Encyclopedia of Philosophy by Jan Sebestik
• Athanasia, in the Internet Archive
• Digitized Bolzano's works

[edit] References

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (January 2008)

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