- Gottlob Frege
Infobox_Philosopher

region = Western Philosophy

era =19th-century philosophy ,

color = #B0C4DE

image_caption = Friedrich Ludwig Gottlob Fregename =

**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

influenced =Giuseppe Peano ,Bertrand Russell ,Rudolf Carnap ,Ludwig Wittgenstein ,Michael Dummett ,Edmund Husserl , and most of the analytic tradition

notable_ideas =Predicate calculus ,Logicism ,Sense and reference **Friedrich Ludwig Gottlob Frege**(8 November 1848 ,Wismar , Grand Duchy ofMecklenburg-Schwerin –26 July 1925 , Bad Kleinen,Germany ) (IPA2|ˈgɔtlop ˈfʁeːgə) was a German mathematician who became alogic ian and philosopher. He helped found both modernmathematical logic andanalytic philosophy . His work had a far-reaching and foundational influence on20th-century philosophy .**Life****Childhood (1848–1869)**Frege was born in 1848 in

Wismar , in the state ofMecklenburg-Schwerin (the modern German federal stateMecklenburg-Vorpommern ). His father, Karl Alexander Frege, was the founder of a girls'high school , of which he was the headmaster until his death in 1866. Afterwards, the school was led by Frege's mother, Auguste Wilhelmine Sophie Frege ("née" Bialloblotzky, apparently of Polish extraction).In childhood, Frege encountered philosophies that would guide his future scientific career. For example, his father wrote a

textbook on the German language for children aged 9-13, the first section of which dealt with the structure andlogic oflanguage .Frege studied at a "gymnasium" in Wismar, and graduated at the age of 15. His teacher

Leo Sachse (also apoet ) played the most important role in determining Frege’s future scientific career, encouraging him to continue his studies at theUniversity of Jena .**tudies at University: Jena and Göttingen (1869 – 1874)**Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the

North German Federation . In the foursemester s of his studies there he attended approximately 20 courses oflectures , most of them onmathematics andphysics . The teacher most important to him wasErnst Abbe (physicist ,mathematician , andinventor ). Abbe gave lectures on "theory ofgravity ", "galvanism andelectrodynamics ", "theory of functions of a complex variable", "applications of physics", "selected divisions ofmechanics ", and "mechanics ofsolids ". Abbe was more than a teacher to Frege: he was a trusted friend, and, as director of the optical manufacturerZeiss , he was in a position to advance Frege's career. After Frege's graduation, they came into closer correspondence.His other notable university teachers were

Karl Snell (subjects: "use ofinfinitesimal analysis ingeometry ", "analytical geometry of planes", "analytical mechanics ", "optics ", "physical foundations of mechanics");Hermann Schäffer ("analytical geometry", "applied physics ", "algebra ic analysis", "on thetelegraph and other electronic machines"); and the famous philosopher,Kuno Fischer ("history of Kantian andcritical philosophy ").Starting in 1871, Frege continued his studies in

Göttingen , the leading university in mathematics in German-speaking territories, where he attended the lectures ofAlfred Clebsch ("analytical geometry "),Ernst Schering ("function theory "),Wilhelm Weber ("physical studies", "applied physics "),Eduard Riecke ("theory ofelectricity "), andRudolf Hermann Lotze ("philosophy of religion "). (Many of the philosophical doctrines of the mature Frege have parallels in Lotze; it has been the subject of scholarly debate whether there was a direct influence arising from Frege's attending Lotze's lectures.)In 1873, Frege attained his

doctorate under Ernst Schering, with adissertation under the title of "Über eine geometrische Darstellung der imaginären Gebilde in der Ebene" ("'On a Geometrical Representation of Imaginary Forms in a Plane"), in which he aimed to solve such fundamental problems in geometry as the mathematical interpretation ofprojective geometry 's infinitely distant (imaginary) points.**Work as a logician**'s theory of truth, is ultimately due to Frege.

One of Frege's stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to "intuition". If there was an intuitive element it was to be isolated and represented separately as an axiom: from there on the proof was to be purely logical and without gaps. Having exhibited this possibility, Frege's more ultimate purpose was to defend the view that

arithmetic is a branch of logic, a view known aslogicism : unlike geometry it was to be shown to have no basis in "intuition," and no need on non-logical axioms. Already in the 1879 "Begriffsschrift" important preliminary theorems, for example a generalized form ofmathematical induction , were derived within what he understood to be pure logic.This idea was formulated in non-symbolic terms in his "Foundations of Arithmetic" of 1884. Later, in the "Basic Laws of Arithmetic" ("Grundgesetze der Arithmetik" (1893, 1903)), published at its author's expense, he attempted to derive all of the laws of arithmetic by use of his symbolism 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")] . The crucial case of the law may be formulated in modern notation as follows. Let {"x"|"Fx"} denote the extension of the predicate "Fx", i.e., the set of all Fs, and similarly for "Gx". Then Basic Law V says that the predicates "Fx" and "Gx" have the same extensioniff ∀x ["Fx" ↔ "Gx"] . The set of Fs is the same as the set of Gs just in case every F is a G and every G is an F. (The case is special because what is here being called the extension of a predicate, or a set, is only one type of "value-range" of a function.)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. It is easy to define the relation of "membership" of a set or extension in Frege's system; Russell then drew attention to "the set of things x that are such that x is not a member of x". The system of the "Grundgesetze" entails both that it is and that it was not a member of itself, and was thus inconsistent. Frege wrote a hasty last-minute appendix to vol. 2, deriving the contradiction and proposing to eliminate it by modifying Basic Law V. (This letter and Frege's reply are translated inJean van Heijenoort 1967.)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:

* Basic Law V can be weakened in other ways. The best-known way is due toGeorge Boolos . A "concept" "F" is "small" if the objects falling under "F" cannot be put in 1-to-1 correspondence with theuniverse of discourse , that is, if: ∃"R" ["R" is 1-to-1 & ∀"x"∃"y"("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"("Fx" ↔ "Gx"). V* is consistent ifsecond-order arithmetic is, and suffices to prove the axioms ofsecond-order arithmetic .

* Basic Law V can simply be replaced withHume's Principle , which says that the number of "F"s is the same as the number of "G"s if and only if the "F"s can be put into a one-to-one correspondence with the "G"s. This principle too is consistent ifsecond-order arithmetic is, and suffices to prove the axioms ofsecond-order arithmetic . This result is termedFrege'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. [*[*]*http://plato.stanford.edu/entries/frege-logic/ Frege's Logic, Theorem, and Foundations for Arithmetic (Stanford Encyclopedia of Philosophy)*] at plato.stanford.edu

* Frege's logic, now known assecond-order logic , can be weakened to so-calledpredicative 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 tomathematical logic was still that ofGeorge Boole and his descendants, especiallyErnst Schroeder . Frege's logical ideas nevertheless spread through the writings of his studentRudolf Carnap and other admirers, particularly Bertrand Russell andLudwig Wittgenstein .**Philosopher**Frege is one of the founders of

analytic philosophy , mainly because of his contributions to thephilosophy of language , including the:

*Function-argument analysis of theproposition ;

*Distinction betweenconcept and object ("Begriff und Gegenstand");

*Principle ofcompositionality ;

*Context principle ;

*Distinction between thesense and reference ("Sinn und Bedeutung") of names and other expressions, sometimes said to involve amediated reference theory .As a philosopher of mathematics, Frege attacked the psychologistic appeal to mental 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 thephilosophy 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 other than "The Foundations of Arithmetic", much of which was mathematical in content, and the first collections of his writings appeared only after World War II. A volume of English translations of Frege's philosophical essays first appeared in 1952, edited by students of Wittgenstein,

Peter Geach andMax Black , with the bibliographic assistance of Wittgenstein (see Geach, ed. 1975, introduction). 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.**"Sinn" and "Bedeutung"**The distinction between "Sinn" and "Bedeutung" (usually translated "Sense and Reference", but also as "Sense and Meaning" or "Sense and Denotation") was an innovation of Frege in his 1892 paper "Über Sinn und Bedeutung" ("On Sense and Reference"). According to Frege, sense and reference are two different aspects of the significance of an expression. Frege applied "Bedeutung" in the first instance to proper names, where it means the bearer of the name, the object in question, but then also to other expressions, including complete sentences, which "bedeuten" the two "truth values", the true and the false; by contrast, the sense or "Sinn" associated with a complete sentence is the thought it expresses. The sense of an expression is said to be the "mode of presentation" of the item referred to. The distinction can be illustrated thus: In their ordinary uses, the name "Charles Philip Arthur George Mountbatten-Windsor," which for logical purposes is an unanalyzable whole, and the functional expression "the Prince of Wales," which contains the significant parts "the prince of ξ" and "Wales", have the same reference, namely the person best known as Prince Charles. But the sense of the word "Wales" is a part of the sense of the latter expression, but no part of the sense of the "full name" of Prince Charles. These distinctions were disputed by Bertrand Russell, especially in his paper "

On Denoting "; the controversy has continued into the present, fueled especially by the famous lectures on "Naming and Necessity " ofSaul Kripke .Imagine the road signs outside a city. They all point to (bedeuten) the same object (the city), although the "mode of presentation" or sense (Sinn) of each sign (its direction or distance) is different. Similarly "the Prince of Wales" and "Charles Philip Arthur George Mountbatten-Windsor" both denote (bedeuten) the same object, though each uses a different "mode of presentation" (sense or Sinn).

**Important dates*** Born

November 8 ,1848 inWismar ,Mecklenburg-Schwerin .

* 1869 — attends theUniversity of Jena .

* 1871 — attends theUniversity of Göttingen .

* 1873 —PhD , doctor inmathematics (geometry ), attained at Göttingen.

* 1874 —Habilitation at Jena; private teacher.

* 1879 — Professor Extraordinarius at Jena.

* 1896 — Ordenlicher Honorarprofessor at Jena.

* 1917 or 1918 — retires.

* DiedJuly 26 ,1925 inBad Kleinen (now part ofMecklenburg-Vorpommern ).**Important works****Logic; foundation of arithmetic**, "eine der arithmetischen nachgebildete Formelsprache des reinen Denkens" (1879). Halle a. S.Begriffsschrift

*English: "Concept Notation, the Formal Language of the Pure Thought like that of Arithmetics".**[**"eine logisch-mathematische Untersuchung über den Begriff der Zahl" (1884). Breslau.*http://www.ac-nancy-metz.fr/enseign/philo/textesph/Frege.pdf Die Grundlagen der Arithmetik:*]

*English: "The Foundations of Arithmetic : the logical-mathematical Investigation of the Concept of Number".**Grundgesetze der Arithmetik**, "Band I" (1893); "Band II" (1903). Jena: Verlag Hermann Pohle.

*English: "Basic Laws of Arithmetic: Vol. 1" (1893); "Vol. 2" (1903).**Philosophical studies****Function and Concept**(1891)

* 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)

* 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)

* Original: " [*http://www.ac-nancy-metz.fr/enseign/philo/textesph/frege_begriff_und_gegenstand.pdf Über Begriff und Gegenstand*] ", in "Vierteljahresschrift für wissenschaftliche Philosophie XVI" (1892): 192-205;

* In English: "Concept and Object ".**What is a Function?**(1904)

* 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 (1918–1923)**Frege intended that the following three papers be published together in a book titled "Logische Untersuchungen (Logical Investigations)". Though the German book never appeared, English translations did appear together in "Logical Investigations", ed. Peter Geach, Blackwells, 1975.

*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.**References****Primary*** [

*http://www.ocf.berkeley.edu/~brianwc/frege/fenglish.html 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 inJean 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 andMax Black , eds., and trans., 1980. "Translations from the Philosophical Writings of Gottlob Frege", 3rd ed. Blackwell (1st ed. 1952).**econdary**"Philosophy":

* 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."

* Diamond, Cora, 1991. "The Realistic Spirit". MIT Press. — Primarily aboutWittgenstein , but contains several articles on Frege.

* Dummett, Michael, 1973. "Frege: Philosophy of Language". Harvard University Press.

* ------, 1981. "The Interpretation of Frege's Philosophy". Harvard University Press.

* Hill, Claire Ortiz, 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.

* Kenny, Anthony, 1995. "Frege — An introduction to the founder of modern analytic philosophy". Penguin Books. — Excellent non-technical introduction and overview of Frege's philosophy.

* Klemke, E.D., ed., 1968. "Essays on Frege". University of Illinois Press. — 31 essays by philosophers, grouped under three headings: 1.Ontology ; 2.Semantics ; and 3.Logic andPhilosophy 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. — On Frege's theory of definitions.

* Sluga, Hans, 1980. "Gottlob Frege". Routledge.

* Smith, Leslie, 1999. "What Piaget Learned from Frege." "Developmental Review 19"(1): 133-153. — On 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."Logic and mathematics":

* Anderson, D. J., andEdward Zalta , 2004, "Frege, Boolos, and Logical Objects," "Journal of Philosophical Logic 33": 1-26.

*Burgess, John, 2005. "Fixing Frege". Princeton Univ. Press. — A critical survey of the ongoing rehabilitation of Frege's logicism.

* Boolos, George, 1998. "Logic, Logic, and Logic". MIT Press. — 12 papers onFrege's theorem and the logicist approach to the foundation ofarithmetic .

* Dummett, Michael, 1991. "Frege: Philosophy of Mathematics". Harvard University Press.

* Demopoulos, William, ed., 1995. "Frege's Philosophy of Mathematics". Harvard Univ. Press. — Papers exploringFrege's theorem and Frege's mathematical and intellectual 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.

* Grattan-Guinness, Ivor, 2000. "The Search for Mathematical Roots 1870-1940". Princeton University Press. — Fair to the mathematician, less so to the philosopher.

* Gillies, Douglas A., 1982. "Frege, Dedekind, and Peano on the foundations of arithmetic". Assen, Netherlands: Van Gorcum.

*Charles Parsons , 1965, "Frege's Concept of Number." Reprinted with Postscript in Demopoulos (1965): 182-210. The starting point of the ongoing sympathetic reexamination of Frege's logicism.

* Wright, Crispin, 1983. "Frege's Conception of Numbers as Objects". Aberdeen University Press. — A systematic exposition and a scope-restricted defense of Frege's "Grundlagen" conception of numbers.**External links*** [

*http://www.ocf.berkeley.edu/~brianwc/frege/ A comprehensive guide to Fregean material available on the web*] by Brian Carver.

*Stanford Encyclopedia of Philosophy :

**" [*http://plato.stanford.edu/entries/frege/ Gottlob Frege*] " — byEdward Zalta .

** " [*http://plato.stanford.edu/entries/frege-logic/ Frege's Logic, Theorem, and Foundations for Arithmetic*] " — by Edward Zalta

*Internet Encyclopedia of Philosophy :

** [*http://www.iep.utm.edu/f/frege.htm Gottlob Frege*] — by Kevin C. Klement.

** [*http://www.utm.edu/research/iep/f/freg-lan.htm Frege and Language*] — by Dorothea Lotter.

*Metaphysics Research Lab: [*http://mally.stanford.edu/frege.html Gottlob Frege.*]

* [*http://www.formalontology.it/fregeg.htm Frege on Being, Existence and Truth.*]

*

* [*http://ctan.org/tex-archive/macros/latex/contrib/begriff/ Begriff,*] aLaTeX package for typesetting Frege's logic notation.Persondata

NAME = Frege, Friedrich Ludwig Gottlob

ALTERNATIVE NAMES = Frege, Gottlob

SHORT DESCRIPTION = Important Germanlogic ian and philosopher

DATE OF BIRTH =November 8 ,1848

PLACE OF BIRTH =Wismar

DATE OF DEATH =July 26 ,1925

PLACE OF DEATH =Bad Kleinen

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