Bernard Bolzano

Bernard (Bernhard) Placidus Johann Nepomuk Bolzano (October 5, 1781(1781-10-05) – December 18, 1848) was a Bohemian mathematician, theologian, philosopher, logician and antimilitarist of German mother tongue. Events 869 - The Fourth Council of Constantinople is convened to decide about what to do about Patriarch Photius of Constantinople Year 1781 ( MDCCLXXXI) was a Common year starting on Monday (link will display the full calendar of the Gregorian calendar (or a Common Events 218 BC - Second Punic War: Battle of the Trebia - Hannibal 's Carthaginian forces defeat those of the Year 1848 ( MDCCCXLVIII) was a Leap year starting on Saturday (link will display the full calendar of the Gregorian Calendar (or a Leap Bohemia (Čechy; Bohemia Czechy is a historical region in central Europe, occupying the western two-thirds of the traditional Czech Lands, currently the A mathematician is a person whose primary area of study and research is the field of Mathematics. Theology is the study of a god or the gods from a religious perspective Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Logic is the study of the principles of valid demonstration and Inference. The German language (de ''Deutsch'') is a West Germanic language and one of the world's major languages.

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. Prague (ˈprɑːg Praha (ˈpraɦa see also other names) is the Capital and Largest city of the Czech Republic. Only two of their twelve children lived to adulthood.

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. Logic is the study of the principles of valid demonstration and Inference. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important 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. Year 1837 ( MDCCCXXXVII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian Calendar (or a Common Philosophy of science is the study of assumptions foundations and implications of Science. Epistemology (from Greek επιστήμη - episteme, "knowledge" + λόγος, " Logos " or theory of knowledge 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. Otto Stolz ( May 3, 1842 – October 25, 1905) was an Austrian mathematician noted for his work on Mathematical analysis and Year 1881 ( MDCCCLXXXI) was a Common year starting on Saturday (link will display the full calendar of the Gregorian calendar (or a Common

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. Year 1837 ( MDCCCXXXVII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian Calendar (or a Common For other uses see Abstract In Philosophy it is commonly considered that every object is either abstract or concrete In Mathematical logic, mereology is a collection of axiomatic First-order theories dealing with parts and their respective wholes 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. Year 1810 ( MDCCCX) was a Common year starting on Monday (link will display the full calendar of the Gregorian calendar (or a Common year 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. Theory of justification is a part of Epistemology that attempts to understand the justification of Propositions and Beliefs Epistemologists are concerned

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. For other uses of Object see Object. In Philosophy, an object is a thing an Entity, or a Being.

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. In Mathematical logic, mereology is a collection of axiomatic First-order theories dealing with parts and their respective 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. In common usage existence is the world of which we are aware through our senses but in Philosophy the word has a more specialized meaning and is often contrasted with 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. In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence An idea is a form (such as a Thought) formed by Consciousness (including Mind) through the Process of ideation. "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. In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield 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. Immanuel Kant (ɪmanuəl kant 22 April 1724 12 February 1804 was an 18th-century German Philosopher from the Prussian city of Königsberg "Noumena" redirects here For the band see Noumena (band.

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. In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true 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. Asymmetric often means simply not symmetric In this sense an asymmetric relation is a Binary relation which is not a Symmetric relation. An explanation is a description which may clarify causes context, and Consequences of a certain object and a phenomenon such as a process, a

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. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides 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). Analysis has its beginnings in the rigorous formulation of Calculus. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true In Mathematical analysis, the intermediate value theorem states that for each value between the upper and lower bounds of the image of a Continuous function In Mathematical analysis, the intermediate value theorem states that for each value between the upper and lower bounds of the image of a Continuous function 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. In Real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician

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. Edmund Gustav Albrecht Husserl (ˈhʊsɛrl April 8 1859 – April 26 1938) was a philosopher, known as the father of Kazimierz Jerzy Skrzypna-Twardowski, Ritter von Ogończyk ( October 20, 1866, Vienna, Austria – February 11, 1938 Franz Clemens Honoratus Hermann Brentano (January 16 1838 &ndash March 17 1917 was an influential German philosopher and psychologist whose influence Through them, and through Gottlob Frege, also an admirer, Bolzano became a formative influence on both phenomenology and analytic philosophy. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Analytic philosophy (sometimes analytical philosophy) is a generic term for a style of Philosophy that came to dominate English-speaking countries in the 20th century

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 
• Biography of Bernard Bolzano
• Bernard Bolzano's Theory of Science
• Bernard Bolzano entry at the Stanford Encyclopedia of Philosophy by Edgar Morscher
• Bolzano's Logic entry at the Stanford Encyclopedia of Philosophy by Jan Sebestik

References

Künne, Wolfgang. The MacTutor History of Mathematics archive is an award-winning website maintained by John J The Stanford Encyclopedia of Philosophy (SEP is a freely-accessible Online encyclopedia of Philosophy maintained by Stanford University. The Stanford Encyclopedia of Philosophy (SEP is a freely-accessible Online encyclopedia of Philosophy maintained by Stanford University. (1998). "Bolzano, Bernard". Routledge Encyclopedia of Philosophy 1: 823-827. London: Routledge. Retrieved on 2007-03-05