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Encyclopedia > Gottlob Frege
Western Philosophy
19th-century philosophy,
Friedrich Ludwig Gottlob Frege
Name: Friedrich Ludwig Gottlob Frege
Birth: November 8, 1848
Death: 26 July 1925
School/tradition: Analytic philosophy
Main interests: Philosophy of mathematics, mathematical logic, Philosophy of language
Notable ideas: Predicate calculus, Logicism, Sense and reference
Influenced: Giuseppe Peano, Bertrand Russell, Rudolf Carnap, Ludwig Wittgenstein, Michael Dummett, Edmund Husserl, and most of the analytic tradition

Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar26 July 1925, IPA: [ˈgɔtlop ˈfʁeːgə]) was a German mathematician who became a logician and philosopher. He helped found both modern mathematical logic and analytic philosophy. His work has had a tremendous influence on 20th-century philosophy, especially in English-speaking countries. In the 18th century the philosophies of The Enlightenment would begin to have dramatic effect, and the landmark works of philosophers such as Immanuel Kant and Jean-Jacques Rousseau would have an electrifying effect on a new generation of thinkers. ... Image File history File links Gottlob_Frege. ... November 8 is the 312th day of the year (313th in leap years) in the Gregorian calendar, with 53 days remaining. ... Year 1848 (MDCCCXLVIII) was a leap year starting on Saturday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Monday of the 12-day slower Julian calendar). ... July 26 is the 207th day (208th in leap years) of the year in the Gregorian calendar, with 158 days remaining. ... 1925 (MCMXXV) was a common year starting on Thursday (link will display the full calendar). ... Analytic philosophy is a generic term for a style of philosophy that came to prominence during the 20th Century. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... Philosophy of language is the reasoned inquiry into the nature, origins, and usage of language. ... First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ... Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. ... The distinction between Sinn and Bedeutung (usually but not always translated sense and reference, respectively) was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper Ãœber Sinn und Bedeutung (On Sense and Reference), which is still widely read today. ... Giuseppe Peano Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ... Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician and advocate for social reform. ... Rudolf Carnap Rudolf Carnap (May 18, 1891, Ronsdorf, Germany – September 14, 1970, Santa Monica, California) was an influential philosopher who was active in central Europe before 1935 and in the United States thereafter. ... Wittgenstein and Hitler in school photograph taken at the Linz Realschule in 1903. ... Sir Michael Anthony Eardley Dummett F.B.A., D. Litt, (born 1925) is a leading British philosopher. ... Edmund Gustav Albrecht Husserl (April 8, 1859, ProstÄ›jov – April 26, 1938, Freiburg) was a German philosopher, known as the father of phenomenology. ... Analytic philosophy is a generic term for a style of philosophy that came to prominence during the 20th Century. ... November 8 is the 312th day of the year (313th in leap years) in the Gregorian calendar, with 53 days remaining. ... Year 1848 (MDCCCXLVIII) was a leap year starting on Saturday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Monday of the 12-day slower Julian calendar). ... Wismar is a small port and Hanseatic League town in northern Germany on the Baltic Sea, in the state of Mecklenburg-Vorpommern, about 45 km due east of Lübeck, and 30 km due north of Schwerin. ... July 26 is the 207th day (208th in leap years) of the year in the Gregorian calendar, with 158 days remaining. ... 1925 (MCMXXV) was a common year starting on Thursday (link will display the full calendar). ... Not to be confused with the NATO phonetic alphabet, which has also informally been called the “International Phonetic Alphabet”. For information on how to read IPA transcriptions of English words, see IPA chart for English. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Logic, from Classical Greek λόγος logos (the word), is the study of the principles and criteria of valid inference and demonstration. ... This article is 58 kilobytes or more in size. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... Analytic philosophy is a generic term for a style of philosophy that came to prominence during the 20th Century. ...

Contents

Life

Childhood (18481869)

Frege was born in 1848 in Wismar, in the state of Mecklenburg-Schwerin (the modern German federal state Mecklenburg-West Pomerania). His father Karl Alexander Frege was the head master and a teacher at a girls' high school until his natural death in 1866. After that, the school was led by Frege's mother, Auguste Wilhelmine Sophie Frege (née Bialloblotzky). His mother in all likelihood had Polish roots. Year 1848 (MDCCCXLVIII) was a leap year starting on Saturday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Monday of the 12-day slower Julian calendar). ... 1869 (MDCCCLXIX) is a common year starting on Friday (link will take you to calendar) of the Gregorian calendar or a common year starting on Sunday of the 12-day-slower Julian calendar. ... Year 1848 (MDCCCXLVIII) was a leap year starting on Saturday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Monday of the 12-day slower Julian calendar). ... Wismar is a small port and Hanseatic League town in northern Germany on the Baltic Sea, in the state of Mecklenburg-Vorpommern, about 45 km due east of Lübeck, and 30 km due north of Schwerin. ... Mecklenburg-Schwerin was a Duchy (from 1815 a Grand Duchy) in northeastern Germany, formed by a partition of the Duchy of Mecklenburg. ... Mecklenburg-Western Pomerania (German: Mecklenburg-Vorpommern) is a Bundesland (federal state) in northern Germany. ... Main article: Secondary education High school is a name used in some parts of the world, and particularly in North America, to describe the last segment of compulsory education. ... In medicine, death by natural causes is a loosely-defined term used by coroners describing death when the cause of death was a naturally occurring disease process, or is not apparent given medical history or circumstances. ... 1866 (MDCCCLXVI) is a common year starting on Monday of the Gregorian calendar or a common year starting on Wednesday of the 12-day-slower Julian calendar. ...


Already in his childhood, Frege met philosophies which would guide his future scientific career. For example, his father wrote a textbook on the German language for 9-13 year olds, and the first section of this book dealt with the structure and logic of language. This article is 58 kilobytes or more in size. ... The examples and perspective in this article or section may not represent a worldwide view. ... Logic, from Classical Greek λόγος logos (the word), is the study of the principles and criteria of valid inference and demonstration. ...


Frege studied at a gymnasium in his home town of Wismar, and graduated at the age of 15, on the Easter of 1869. With reference to his future scientific career, his teacher, Leo Sachse (also a poet, who gave his name to a street in Jena) played the most important role, encouraging him to continue his studies at the University of Jena. A gymnasium (pronounced with or, in Swedish, as opposed to ) is a type of school providing secondary education in some parts of Europe, comparable to English Grammar Schools and U.S. High Schools. ... 1869 (MDCCCLXIX) is a common year starting on Friday (link will take you to calendar) of the Gregorian calendar or a common year starting on Sunday of the 12-day-slower Julian calendar. ... The poor poet A poet is a person who writes poetry. ... The Market Square in Jena. ... Friedrich Schiller University of Jena (FSU) is located in Jena, Thuringia in Germany and was named for the German writer Friedrich Schiller. ...


Studies at University: Jena and Göttingen (18691874)

Frege signed up to the University of Jena in the spring of 1869 as a citizen of the North German Federation. In the 4 semesters of his studies here he attended around 20 lectures, primarily on mathematics and physics. The progress he made in his studies was excellent. Göttingen ( ) is a city in Lower Saxony, Germany. ... 1869 (MDCCCLXIX) is a common year starting on Friday (link will take you to calendar) of the Gregorian calendar or a common year starting on Sunday of the 12-day-slower Julian calendar. ... Year 1874 (MDCCCLXXIV) was a common year starting on Thursday (link with display the full calendar) of the Gregorian calendar (or a common year starting on Saturday of the 12-day slower Julian calendar). ... Map of the North German Confederation Capital Berlin Political structure Confederation Presidency Prussia (William I) Chancellor Otto von Bismarck History  - Constitution tabelled April 16, 1867  - Confederation formed July 1, 1867  - Elevation to empire January 18, 1871 The North German Federation (in German, Norddeutscher Bund) came into existence in 1867, following... An academic term is the time during which a school, college or university holds classes. ... A lecture is a talk on a particular subject given in order to teach people about that subject, for example by a university or college teacher. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe. ...


His most important teacher was Ernst Abbe (physicist, mathematician and inventor). Abbe gave Frege lectures on The Theory of Gravity, Galvanism and electrodynamics, The theory of functions of a complex variable, Applications of physics, Selected divisions of mechanics, and The mechanics of solids. Abbe, not as a teacher, but as director of Zeiss, the optical manufacturers, and as a trusted friend had a great effect on Frege, and after Frege's (absolution?) they came into closer correspondence. Ernst Karl Abbe Ernst Karl Abbe (January 23, 1840 in Eisenach – January 14, 1905 in Jena), was a German physicist. ... ... Leonhard Euler is considered by many to be one of the greatest mathematicians of all time A mathematician is the person whose primary area of study and research is the field of mathematics. ... For other uses, see Inventor (disambiguation). ... Gravity is a force of attraction that acts between bodies that have mass. ... In biology, galvanism is the contraction of a muscle that is stimulated by an electric current. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ... A solid is a state of matter, characterized by a definite volume and a definite shape (i. ... Carl Zeiss The Carl Zeiss company is a German manufacturer of optical systems, industrial measurements and medical devices originally founded in Jena in 1846 by Carl Zeiss, Ernst Abbe and Otto Schott. ...


His other notable university teachers were Karl Snell (subjects: The use of infinitesimal analysis in geometry, The analytical geometry of planes, Analytical mechanics, Optics, The physical foundations of mechanics); Hermann Schäffer (Analytical geometry, Applied physics, Algebraic analysis, On the telegraph and other electronic machines); and a famous philosopher, Kuno Fischer (The history of Kantian and critical philosophy). In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... Table of Geometry, from the 1728 Cyclopaedia. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ... In mathematics, a plane is the fundamental two-dimensional object. ... Analytical mechanics is a term used for a refined, highly mathematical form of classical mechanics, constructed from the eighteenth century onwards as a formulation of the subject as founded by Isaac Newton. ... Table of Opticks, 1728 Cyclopaedia Optics ( appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... Cutout of the ITER project Applied physics is physics that is intended for a particular technological or practical use, as for example in engineering, as opposed to basic research. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Telegraphy (from the Greek words tele = far away and grapho = write) is the long distance transmission of written messages without physical transport of letters, originally over wire. ... // Electronics is the study of electron mechanics. ... Kuno Fischer, born Ernst Kuno Berthold Fischer, (July 23, 1824 - July 5, 1907) was a German philosopher. ... Kantianism is the philosophy of Immanuel Kant. ... Attributed to Immanuel Kant, the critical philosophy movement sees the primary task of philosophy as criticism rather than justification. ...


In 1871, Frege continued his studies in Göttingen, the leading university in mathematics in German-speaking territories. Here, he attended the lectures of Alfred Clebsch (Analytical geometry), Ernst Schering (Function theory), Wilhelm Weber (Physical studies, Applied physics), Eduard Riecke (The theory of electricity) and (in the worlds of Werner Stelzner), "ingenious philosopher" Rudolf Hermann Lotze (The philosophy of religion). In many aspects, the ideologies of Frege and Lotze agree: in the philosophy of Frege, there are many items which point to Lotze's influence (for example, they both expressed strong opposition to one of the era's new philosophical sciences, psychology), and it has been the object of many debates whether he gained these ideas in his time at Göttingen and primarily due to Lotze: this is not for sure. 1871 (MDCCCLXXI) was a common year starting on Sunday (see link for calendar). ... Göttingen ( ) is a city in Lower Saxony, Germany. ... Alfred Clebsch (1832-1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ... Ernst Christian Friedrich Schering (1824-1889) Ernst Christian Friedrich Schering (May 31, 1824 – December 27, 1889) was a German apothecary and industrialist who created the Schering Corporation. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... Wilhelm Eduard Weber (October 24, 1804 - June 23, 1891) was a noted physicist. ... Cutout of the ITER project Applied physics is physics that is intended for a particular technological or practical use, as for example in engineering, as opposed to basic research. ... Lightning strikes during a night-time thunderstorm. ... Rudolf Herman Lotze (May 21, 1817 - July 1, 1881), was a German philosopher. ... Philosophy of religion is the rational study of the meaning and justification ( or rebuttal) of fundamental religious claims, particularly about the nature and existence of God (or gods, or the divine). ... Psychology is an academic or applied discipline involving the scientific study of mental processes such as perception, cognition, emotion, personality, behavior, and interpersonal relationships. ...


In 1873 Frege attained his doctorate with Ernst Schering, with a dissertation under the title of "Über eine geometrische Darstellung der imagiäre Gebilde in der Ebene" ("The geometrical and planar portrayal of imaginary shapes), in which he aimed to solve such fundamental problems in geometry as the mathematical interpretation of projective geometry's infinitely distant (imaginary) points. 1873 (MDCCCLXXIII) was a common year starting on Wednesday (see link for calendar). ... This article is about the thesis in dialectics and academia. ... Something is called planar if it is made up of flat planes, or pertaining to planes. ... Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ...


Work as a Logician

Main article: Begriffsschrift

Though his education and early work were mathematical, and especially geometrical, Frege's thought soon turned to logic. He is fact widely regarded as a logician on a par with Aristotle. His 1879 Begriffsschrift (Concept Script) marked a turning point in the history of logic. The Begriffsschrift broke much new ground, including a clean treatment of functions and variables. Frege wanted to show that mathematics grew out of logic, but in so doing devised techniques that took him far beyond the Aristotelian syllogistic and Stoic propositional logic that had come down to him in the logical tradition. In effect, he invented axiomatic predicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and solved the problem of multiple generality. Though previous logic had dealt with the logical constants and, or, if...then..., not, and some and all, iterations of these operations were little understood; even the distinction between a pair of sentences like "every boy loves some girl" and "some girl is loved by every boy" could not be represented. It is sometimes noted that Aristotle's logic would not be able to represent even the most elementary inferences in Euclid's geometry, but Frege's "conceptual notation" could represent inferences involving indefinitely complex mathematical statements. Hence the analysis of logical concepts and the machinery of formalization that is essential to Bertrand Russell's theory of descriptions and Principia Mathematica (with Alfred North Whitehead), and to Gödel's incompleteness theorems, and to Alfred Tarski's theory of truth, is ultimately due to Frege. Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. ... Aristotle (Greek: AristotélÄ“s) (384 BC – March 7, 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. ... Partial plot of a function f. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... Logic, from Classical Greek λόγος logos (the word), is the study of the principles and criteria of valid inference and demonstration. ... In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. ... ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The problem of multiple generality names a failure in Aristotelian logic to describe certain intuitively valid inferences. ... In symbolic logic, a logical constant is a symbol that has the same semantic value in all models. ... Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician and advocate for social reform. ... The theory of descriptions is one of the philosopher Bertrand Russells most significant contributions to the philosophy of language. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ... Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England – December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ... [...]I dont believe in natural science. ... In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ... // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ...


Frege's purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism. Already in the 1879 Begriffschrifft important preliminary theorems related to mathematical induction were derived within pure logic. Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...


In his later Grundgesetze der Arithmetik (1893, 1903), published at its author's expense, he attempted to derive all of the laws of arithmetic from axioms he asserted as logical. Most of these axioms were carried over from his Begriffsschrift, though not without some significant changes. The one truly new principle was one he called the Basic Law V: the "value-range" of the function f(x) is the same as the "value-range" of the function g(x) if and only if ∀x[f(x) = g(x)]. In modern notation and terminology, let {x|Fx} denote the extension of the predicate Fx, and similarly for Gx. Then Basic Law V says that the predicates Fx and Gx have the same extension iff ∀x[FxGx]. Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. ... In metaphysics, extension is the property of taking up space; see Extension (metaphysics). ... In linguistics and logic, a predicate is an expression that can be true of something. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...


In a famous episode, Bertrand Russell wrote to Frege, just as Vol. 2 of the Grundgesetze was about to go to press in 1903, showing that Russell's paradox could be derived from Frege's Basic Law V. (This letter and Frege's reply thereto are translated in Jean van Heijenoort 1967.) Hence the system of the Grundgesetze was inconsistent. Frege wrote a hasty last-minute appendix to vol. 2, deriving the contradiction and proposing to eliminate it by modifying Basic Law V. Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician and advocate for social reform. ... (For E. W. Russells Paradox, see Religious and militarist attitudes and Paradox supported. ... Jean van Heijenoort (prounounced highenort) (July 23, 1912, Creil France - March 29, 1986, Mexico City) was a pioneer historian of mathematical logic. ...


Frege's proposed remedy was subsequently shown to imply that there is but one object in the universe of discourse, and hence is worthless (indeed this would make for a contradiction in Frege's system if he had axiomatized the idea, fundamental to his discussion, that the True and the False are distinct objects; see e.g. Dummett 1973). But recent work has shown that much of the program of the Grundgesetze might be salvaged in other ways: The term universe of discourse generally refers to the entire set of terms used in a specific discourse, i. ... Sir Michael Anthony Eardley Dummett F.B.A., D. Litt, (born 1925) is a leading British philosopher. ...

  • Basic Law V can be weakened in other ways. The best-known way is due to George Boolos. A "concept" F is "small" if the objects falling under F cannot be put in 1-to-1 correspondence with the universe of discourse, that is, if: ¬∃R[R is 1-to-1 & ∀xy(xRy & Fy)]. Now weaken V to V*: a "concept" F and a "concept" G have the same "extension" if and only if neither F nor G is small or ∀x(FxGx). V* is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic.
  • Basic Law V can simply be replaced with Hume's Principle, which says that the number of Fs is the same as the number of Gs if and only if the Fs can be put into a one-to-one correspondence with the Gs. This principle too is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic. This result is termed Frege's Theorem because it was noticed that in developing arithmetic, Frege's use of Basic Law V is restricted to a proof of Hume's Principle; it is from this in turn that arithmetical principles are derived. On Hume's Principle and Frege's Theorem, see [1].
  • Frege's logic, now known as second-order logic, can be weakened to so-called predicative second-order logic. However, this logic, although provably consistent by finitistic or constructive methods, can interpret only very weak fragments of arithmetic.

Frege's work in logic was little recognized in his day, in considerable part because his peculiar diagrammatic notation had no antecedents; it has since had no imitators. Moreover, until Principia Mathematica appeared, 1910-13, the dominant approach to mathematical logic was still that of George Boole and his descendants, especially Ernst Schroeder. Frege's logical ideas nevertheless spread through the writings of his student Rudolph Carnap and other admirers, particularly Russell and Wittgenstein. George Stephen Boolos (September 4, 1940, New York City - May 27, 1996) was a philosopher and a mathematical logician. ... The term universe of discourse generally refers to the entire set of terms used in a specific discourse, i. ... In mathematical logic, second order arithmetic is a stronger version of Peano arithmetic that allows quantification over subsets of the integers, rather than just over integers. ... In mathematical logic, second order arithmetic is a stronger version of Peano arithmetic that allows quantification over subsets of the integers, rather than just over integers. ... Humes principle is a standard for comparing any two sets of objects as to size. ... In mathematical logic, second order arithmetic is a stronger version of Peano arithmetic that allows quantification over subsets of the integers, rather than just over integers. ... In mathematical logic, second order arithmetic is a stronger version of Peano arithmetic that allows quantification over subsets of the integers, rather than just over integers. ... Freges theorem states that the axioms of second-order arithmetic can be derived in second-order logic from Humes principle. ... In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ... In mathematics, a predicate is a relation. ... In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... George Boole [], (November 2, 1815 – December 8, 1864) was a British mathematician and philosopher. ... Ernst Schröder (25 November 1841 - 16 June 1902) was the most significant representative of the algebraic logic school in Germany in the second half of the nineteenth century. ... Rudolf Carnap (May 18, 1891 - September 14, 1970) was a German philosopher. ...


Philosopher

Frege is one of the founders of analytic philosophy, mainly because of his contributions to the philosophy of language, including the: The distinction between Sinn and Bedeutung (usually but not always translated sense and reference, respectively) was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper Ãœber Sinn und Bedeutung (On Sense and Reference), which is still widely read today. ... Analytic philosophy is a generic term for a style of philosophy that came to prominence during the 20th Century. ... Philosophy of language is the reasoned inquiry into the nature, origins, and usage of language. ...

As a philosopher of mathematics, Frege attacked the psychologistic appeal to "mental", or "inner" explanations of the content of judgment of the meaning of sentences. His original purpose was very far from answering general questions about meaning; instead, he devised his logic to explore the foundations of arithmetic, undertaking to answer questions such as "What is a number?" or "What objects do number-words ("one", "two", etc.) refer to?" But in pursuing these matters, he eventually found himself analysing and explaining what meaning is, and thus came to several conclusions that proved highly consequential for the subsequent course of analytic philosophy and the philosophy of language. Look up Function in Wiktionary, the free dictionary. ... Proposition is a term used in logic to describe the content of assertions. ... The distinction between Sinn and Bedeutung (usually but not always translated sense and reference, respectively) was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper Ãœber Sinn und Bedeutung (On Sense and Reference), which is still widely read today. ... The mediated reference theory is a semantic theory that posits that words refer to something in the external world, but insists that there is more to the meaning of a name than simply the object to which it refers. ... The distinction between concept and object is due to the German philosopher Gottlob Frege. ... In the philosophy of language, the context principle is a form of semantic holism holding that a philosopher should never . ... The Principle of Compositionality in semantics is the principle that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them. ... Psychologism in the philosophy of mathematics is the explanation or derivation of mathematical or logical laws in terms of psychological facts. ... Analytic philosophy is a generic term for a style of philosophy that came to prominence during the 20th Century. ... Philosophy of language is the reasoned inquiry into the nature, origins, and usage of language. ...


It should be kept in mind that Frege was employed as a mathematician, not a philosopher, and published his philosophical papers in scholarly journals that often were hard to access outside of the German speaking world. He never published a philosophical monograph and the first collections of his writings appeared only after WWII. A translation of Frege's philosophical essays first appeared in English Hence despite the generous praise of Russell and Wittgenstein, Frege was little known as a philosopher during his lifetime. His ideas spread chiefly through those he influenced, such as Russell, Wittgenstein, and Carnap, and through Polish work on logic and semantics. Russell is an Irish/or French name that means anything that is or relates to the colour red or a fox. ... Ludwig Wittgenstein (1889-1951), pictured here in 1930, made influential contributions to Logic and the philosophy of language, critically examining the task of conventional philosophy and its relation to the nature of language. ... Ludwig Wittgenstein (1889-1951), pictured here in 1930, made influential contributions to Logic and the philosophy of language, critically examining the task of conventional philosophy and its relation to the nature of language. ... Rudolf Carnap (May 18, 1891 - September 14, 1970) was a German philosopher. ...


Thought

A great deal of Frege's writings were not translated into English until the 1950s. "Thought is one of those writings. It was published in 1918 as the first part of a series of three papers entitled Logical Investigations. Along with On Sense and Reference, it is one of Frege's most influential and widely discussed papers" (Beaney, 1997).


Sinn and Bedeutung

The distinction between Sinn and Bedeutung (usually but not always translated Sense and Reference) was an innovation of Frege in his 1892 paper Über Sinn und Bedeutung (On Sense and Reference), which is still widely read today. According to Frege, sense and reference are two different aspects of the meaning of at least some kinds of terms (Frege applied "Bedeutung" mainly to proper names and, to a lesser extent, sentences). Roughly, a term's reference (sometimes, "referent") is the object it refers to and its sense is how it presents that object. The distinction can be illustrated thus: in their ordinary uses, the name "Charles Philip Arthur George Mountbatten-Windsor" and the functional expression "the Prince of Wales" differ in sense, but have the same reference. If someone else is the Prince of Wales in 2050, then the 2050 use and the present use of "the Prince of Wales" differ in reference but may have the same sense. This is an oversimplification, however: for one thing, the sentence is the unit of meaning for Frege, and consequently expressions have sense or reference only in contexts of use. Other complications arise about the sense and reference of expressions in certain contexts, such as expressions of belief. The distinction between Sinn and Bedeutung (usually but not always translated sense and reference, respectively) was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper Ãœber Sinn und Bedeutung (On Sense and Reference), which is still widely read today. ...


The young Frege (18741884)

Translation forthcoming from Hungarian.

In 1874 Frege returned to Jena, and under Ernst Haeckel's deanship. Year 1874 (MDCCCLXXIV) was a common year starting on Thursday (link with display the full calendar) of the Gregorian calendar (or a common year starting on Saturday of the 12-day slower Julian calendar). ... 1884 (MDCCCLXXXIV) is a leap year starting on Tuesday (click on link to calendar) of the Gregorian calendar (or a leap year starting on Thursday of the 12-day-slower Julian calendar). ... Ernst Haeckel. ...


The mature Frege (18931906)

Translation forthcoming from Hungarian.

Year 1893 (MDCCCXCIII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Tuesday of the 12-day slower Julian calendar). ... 1906 (MCMVI) was a common year starting on Monday (see link for calendar). ...

The old Frege (19061923)

Translation forthcoming from Hungarian.

1906 (MCMVI) was a common year starting on Monday (see link for calendar). ... {{year nav|1939 1923 (MCMXXIII) was a common year starting on Monday (link will display the full calendar). ...

Important dates

November 8 is the 312th day of the year (313th in leap years) in the Gregorian calendar, with 53 days remaining. ... Year 1848 (MDCCCXLVIII) was a leap year starting on Saturday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Monday of the 12-day slower Julian calendar). ... Wismar is a small port and Hanseatic League town in northern Germany on the Baltic Sea, in the state of Mecklenburg-Vorpommern, about 45 km due east of Lübeck, and 30 km due north of Schwerin. ... Mecklenburg-Schwerin was a Duchy (from 1815 a Grand Duchy) in northeastern Germany, formed by a partition of the Duchy of Mecklenburg. ... 1869 (MDCCCLXIX) is a common year starting on Friday (link will take you to calendar) of the Gregorian calendar or a common year starting on Sunday of the 12-day-slower Julian calendar. ... Friedrich Schiller University of Jena (FSU) is located in Jena, Thuringia in Germany and was named for the German writer Friedrich Schiller. ... 1871 (MDCCCLXXI) was a common year starting on Sunday (see link for calendar). ... The Georg-August University of Göttingen (Georg-August-Universität Göttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ... 1873 (MDCCCLXXIII) was a common year starting on Wednesday (see link for calendar). ... PhD usually refers to the academic title Doctor of Philosophy PhD can also refer to the manga Phantasy Degree This is a disambiguation page — a list of pages that otherwise might share the same title. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Table of Geometry, from the 1728 Cyclopaedia. ... Year 1874 (MDCCCLXXIV) was a common year starting on Thursday (link with display the full calendar) of the Gregorian calendar (or a common year starting on Saturday of the 12-day slower Julian calendar). ... Habilitation is the highest academic degree a person can achieve by his/her own pursuit. ... 1879 (MDCCCLXXIX) was a common year starting on Wednesday (see link for calendar). ... Year 1896 (MDCCCXCVI) was a leap year starting on Wednesday (link will display calendar). ... Year 1917 (MCMXVII) was a common year starting on Monday of the Gregorian calendar (see link for calendar) or a common year starting on Tuesday of the 13-day slower Julian calendar (see: 1917 Julian calendar). ... 1918 (MCMXVIII) was a common year starting on Tuesday of the Gregorian calendar (see link for calendar) or a common year starting on Wednesday of the Julian calendar. ... July 26 is the 207th day (208th in leap years) of the year in the Gregorian calendar, with 158 days remaining. ... 1925 (MCMXXV) was a common year starting on Thursday (link will display the full calendar). ... Mecklenburg-Western Pomerania (German: Mecklenburg-Vorpommern) is a Bundesland (federal state) in northern Germany. ...

Important Works

First-order logic and foundations of arithmetic

Begriffsschrift (1879) First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... 1879 (MDCCCLXXIX) was a common year starting on Wednesday (see link for calendar). ...

  • Original: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle a. S., 1879;
  • In English: Concept Notation, the Formal Language of the Pure Thought like that of Arithmetics).

The Foundations of Arithmetics (1884) Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. ... 1884 (MDCCCLXXXIV) is a leap year starting on Tuesday (click on link to calendar) of the Gregorian calendar (or a leap year starting on Thursday of the 12-day-slower Julian calendar). ...

  • Original: Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl; Breslau, 1884;
  • In English: The Foundations of Arithmetics: the logical-mathematical investigation of the Concept of Number.

Basic Laws of Arithmetic, Vol. 1 (1893); Vol. 2 (1903) Year 1893 (MDCCCXCIII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Tuesday of the 12-day slower Julian calendar). ... 1900 (MCMIII) was a common year starting on Thursday (link will display calendar) of the Gregorian calendar or a common year starting on Friday of the 13-day slower Julian calendar. ...

  • Original: Grundgesetze der Arithmetik, Jena: Verlag Hermann Pohle, Band I (1893), Band II (1903);
  • In English: Basic Laws of Arithmetic.

Philosophical studies

Function and Concept (1891) Year 1891 (MDCCCXCI) was a common year starting on Thursday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Saturday of the 12-day slower Julian calendar). ...

  • Original: Funktion und Begriff : Vortrag, gehalten in der Sitzung; vom 9. Januar 1891 der Jenaischen Gesellschaft für Medizin und Naturwissenschaft, Jena, 1891;
  • In English: Function and Concept.

On Sense and Reference (1892) On Function and Concept (Ãœber Funktion und Begriff) is an article by Gottlob Frege, published in 1891. ... 1892 (MDCCCXCII) was a leap year starting on Friday (see link for calendar). ...

  • Original: Über Sinn und Bedeutung; in Zeitschrift für Philosophie und philosophische Kritik C (1892): 25-50;
  • In English: On Sense and Reference.

Concept and Object (1892) The distinction between Sinn and Bedeutung (usually but not always translated sense and reference, respectively) was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper Ãœber Sinn und Bedeutung (On Sense and Reference), which is still widely read today. ... 1892 (MDCCCXCII) was a leap year starting on Friday (see link for calendar). ...

  • Original: Über Begriff und Gegenstand, in Vierteljahresschrift für wissenschaftliche Philosophie XVI (1892): 192-205;
  • In English: Concept and Object.

What is a Function? (1904) The distinction between concept and object is due to the German philosopher Gottlob Frege. ... 1904 (MCMIV) was a leap year starting on a Friday (see link for calendar). ...

  • Original (in German): Was ist eine Funktion?, in Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20. Februar 1904, S. Meyer (ed.), Leipzig, 1904, pp. 656-666;
  • In English: What is a Function?

Logical Investigations (19181923) Frege intended that the following three papers be published together in a book titled Logische Untersuchungen (Logical Investigations). English translations did appear together in 1975. 1918 (MCMXVIII) was a common year starting on Tuesday of the Gregorian calendar (see link for calendar) or a common year starting on Wednesday of the Julian calendar. ... {{year nav|1939 1923 (MCMXXIII) was a common year starting on Monday (link will display the full calendar). ...

  • 1918-19. "Der Gedanke: Eine logische Untersuchung (Thought: A Logical Investigation)" in Beiträge zur Philosophie des Deutschen Idealismus I: 58-77.
  • 1918-19. "Die Verneinung" (Negation)" in Beiträge zur Philosophie des deutschen Idealismus I: 143-157.
  • 1923. "Gedankengefüge (Compound Thought)" in Beiträge zur Philosophie des Deutschen Idealismus III: 36-51.

Articles on Geometry

  • 1903: Über die Grundlagen der Geometrie. II. Jaresbericht der deutschen Mathematiker-Vereinigung XII (1903), 368-375;
    • In English: On the Foundations of Geometry.
  • 1967: Kleine Schriften. (I. Angelelli, ed.) Wissenschaftliche Buchgesellschaft. Darmstadt, 1967 és G. Olms, Hildescheim, 1967. "Small Writings", a collection of most of his writings (e.g. the previous), posthumously published.

1900 (MCMIII) was a common year starting on Thursday (link will display calendar) of the Gregorian calendar or a common year starting on Friday of the 13-day slower Julian calendar. ... 1967 (MCMLXVII) was a common year starting on Sunday of the Gregorian calendar (the link is to a full 1967 calendar). ... A posthumous work is a work of art (generally a book, musical composition, musical recording, or movie) that is published after the death of an author or performer. ...

References

Primary

  • Online bibliography of Frege's works and their English translations.
  • 1879. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: Louis Nebert. Translation: Concept Script, a formal language of pure thought modelled upon that of arithmetic, by S. Bauer-Mengelberg in Jean Van Heijenoort, ed., 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press.
  • 1884. Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. Translation: J. L. Austin, 1974. The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number, 2nd ed. Blackwell.
  • 1891. "Funktion und Begriff." Translation: "Function and Concept" in Geach and Black (1980).
  • 1892a. "Über Sinn und Bedeutung" in Zeitschrift für Philosophie und philosophische Kritik 100: 25-50. Translation: "On Sense and Reference" in Geach and Black (1980).
  • 1892b. "Über Begriff und Gegenstand" in Vierteljahresschrift für wissenschaftliche Philosophie 16: 192-205. Translation: "Concept and Object" in Geach and Black (1980).
  • 1893. Grundgesetze der Arithmetik, Band I. Jena: Verlag Hermann Pohle. Band II, 1903. Partial translation: Furth, M, 1964. The Basic Laws of Arithmetic. Uni. of California Press.
  • 1904. "Was ist eine Funktion?" in Meyer, S., ed., 1904. Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20. Februar 1904. Leipzig: Barth: 656-666. Translation: "What is a Function?" in Geach and Black (1980).
  • Peter Geach and Max Black, eds., and trans., 1980. Translations from the Philosophical Writings of Gottlob Frege, 3rd ed. Blackwell.

Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. ... Jean van Heijenoort (prounounced highenort) (July 23, 1912, Creil France - March 29, 1986, Mexico City) was a pioneer historian of mathematical logic. ... John Langshaw Austin (March 28, 1911 - February 8, 1960) was a philosopher of language, who developed much of the current theory of speech acts. ... The distinction between Sinn and Bedeutung (usually but not always translated sense and reference, respectively) was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper Über Sinn und Bedeutung (On Sense and Reference), which is still widely read today. ... The distinction between Sinn and Bedeutung (usually but not always translated sense and reference, respectively) was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper Über Sinn und Bedeutung (On Sense and Reference), which is still widely read today. ... Peter Thomas Geach (born 1919) is one of the foremost contemporary British philosophers. ... Max Black (24 February 1909, Baku, Russian Empire [present-day Azerbaijan] – 27 August 1988, Ithaca, New York, United States) was a distinguished Anglo-American philosopher, who was a leading influence in analytic philosophy in the first half of the twentieth century. ...

Secondary

  • Anderson, D. J., and Edward Zalta, 2004, "Frege, Boolos, and Logical Objects," Journal of Philosophical Logic 33: 1-26.
  • Baker, Gordon, and P.M.S. Hacker, 1984. Frege: Logical Excavations. Oxford University Press.
Vigorous, if controversial, criticism of both Frege's philosophy and influential contemporary interpretations such as Dummett's.
  • Burgess, John, 2005. Fixing Frege. Princeton Univ. Press.
A critical survey of the work by Boolos, Heck, and others attempting to rehabilitate Frege's logicism.
Contains 12 papers on Frege's logic and logistic approach to the foundations of arithmetic.
  • Diamond, Cora, 1991. The Realistic Spirit. MIT Press.
Ostensibly about Wittgenstein, but contains several valuable articles on Frege.
  • Michael Dummett, 1973. Frege: Philosophy of Language. Harvard University Press.
  • ------, 1981. The Interpretation of Frege's Philosophy. Harvard University Press.
  • ------, 1991. Frege: Philosophy of Mathematics. Harvard University Press.
  • Demopoulos, William, 1995. "Frege's Philosophy of Mathematics". Harvard Univ. Press.
Explores the significance of Frege's theorem, and his mathematical and intellectural background.
  • Ferreira, F. and Wehmeier, K., 2002, "On the consistency of the Delta-1-1-CA fragment of Frege's Grundgesetze," Journal of Philosophic Logic 31: 301-11.
  • Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton University Press.
Fair to the mathematician, less so to the philosopher.
  • Douglas A. Gillies, 1982. Frege, Dedekind, and Peano on the foundations of arithmetic. Assen, Netherlands: Van Gorcum.
  • Hatcher, William, 1982. The Logical Foundations of Mathematics. Pergamon.
Chpt. 3 recasts the system of the Grundgesetze in modern notation, and derives the Peano axioms in this system using natural deduction.
  • Hill, C. O., 1991. Word and Object in Husserl, Frege and Russell: The Roots of Twentieth-Century Philosophy. Athens OH: Ohio University Press.
  • ------, and Rosado Haddock, G. E., 2000. Husserl or Frege: Meaning, Objectivity, and Mathematics. Open Court.
On the Frege-Husserl-Cantor triangle.
  • Klemke, E.D., ed., 1968. Essays on Frege. University of Illinois Press.
Contains a total of thirty-one essays on Frege's work by prominent philosophers; essays divided into three part subject matter sections: 1. 'Frege's Ontology', 2. 'Frege's Semantics', and 3. 'Frege's Logic and Philosophy of Mathematics'.
  • Rosado Haddock, Guillermo E., 2006. A Critical Introduction to the Philosophy of Gottlob Frege. Ashgate Publishing.
  • Sisti, Nicola, 2005. Il Programma Logicista di Frege e il Tema delle Definizioni. Franco Angeli.
Analyses and explains Frege's thought on definitions.
  • Hans Sluga, 1980. Gottlob Frege. Routledge.
  • Smith, Leslie, 1999. "What Piaget Learned from Frege." Developmental Review 19(1): 133-153.
An examination of why Frege first appears in Piaget's writings in 1949, twenty-five years after he began publishing on logic and epistemology.
  • Weiner, Joan, 1990. Frege in Perspective. Cornell University Press.
  • Crispin Wright, 1983. Frege's Conception of Numbers as Objects. Aberdeen University Press.
Written from the viewpoint of a modern philosopher of language and logic, contains a systematic exposition and a scope-restricted defense of Frege's Grundlagen conception of numbers.

Edward N. Zalta is a Senior Research Scholar at the Center for the Study of Language and Information. ... George Stephen Boolos (September 4, 1940, New York City - May 27, 1996) was a philosopher and a mathematical logician. ... Sir Michael Anthony Eardley Dummett F.B.A., D. Litt, (born 1925) is a leading British philosopher. ... Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ... In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic. ... In mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it naturally occurs. ... Hans Sluga is Professor of Philosophy at the University of California, Berkeley. ... Jean Piaget [] (August 9, 1896 – September 16, 1980) was a Swiss philosopher, natural scientist and developmental psychologist, well known for his work studying children and his theory of cognitive development. ... Crispin Wright (born 1942) is a British philosopher, who has written on neo-Fregean philosophy of mathematics, Wittgensteins later philosophy, and on issues related to truth, realism, cognitivism, skepticism, knowledge, and objectivity. ...

External links

Wikisource
Wikisource has original works written by or about:
Gottlob Frege
Philosophy Portal

Image File history File links Wikisource-logo. ... The original Wikisource logo. ... The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ... Edward N. Zalta is a Senior Research Scholar at the Center for the Study of Language and Information. ... The Internet Encyclopedia of Philosophy is an online encyclopedia on philosophical topics and philosophers founded by James Fieser in 1995. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... The LaTeX logo, typeset with LaTeX LATEX, written as LaTeX in plain text, is a document markup language and document preparation system for the TeX typesetting program. ... Image File history File links Socrates. ...


  Results from FactBites:
 
Gottlob Frege - Wikipedia, the free encyclopedia (1043 words)
Frege was a major advocate of the view that arithmetic is reducible to logic, a form of logicism.
Frege never succeeded in amending his axioms to his satisfaction, although Russell's theory of types, the axiomatic set theory of Ernst Zermelo and John von Neumann, and the second order logic of George Boolos (1998) all suggested ways to remedy the problem.
Frege is regarded as one of the founding fathers of analytic philosophy, mainly because of his conceptual contributions to the philosophy of language, such as his:
  More results at FactBites »

 
 

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