Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German

online version

. * In English: ''Begriffsschrift, a Formula Language, Modeled Upon That of Arithmetic, for Pure Thought'', in: J. van Heijenoort (ed.), ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931'', Harvard, MA: Harvard University Press, 1967, pp. 5–82. * In English (selected sections revised in modern formal notation): R. L. Mendelsohn, ''The Philosophy of Gottlob Frege'', Cambridge: Cambridge University Press, 2005: "Appendix A. Begriffsschrift in Modern Notation: (1) to (51)" and "Appendix B. Begriffsschrift in Modern Notation: (52) to (68)." '' Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl'' (1884), Breslau: Verlag von Wilhelm Koebner

online version

. * In English: '' The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number'', translated by J. L. Austin, Oxford: Basil Blackwell, 1950. ''Grundgesetze der Arithmetik'', Band I (1893); Band II (1903), Jena: Verlag Hermann Pohle

online version)

* In English (translation of selected sections), "Translation of Part of Frege's ''Grundgesetze der Arithmetik''," translated and edited Peter Geach and

Band I

an

Band II

. * In German (revised in modern formal notation): ''Grundgesetze der Arithmetik – Begriffsschriftlich abgeleitet. Band I und II: In moderne Formelnotation transkribiert und mit einem ausführlichen Sachregister versehen'', edited by T. Müller, B. Schröder, and R. Stuhlmann-Laeisz, Paderborn: mentis, 2009. * In English: ''Basic Laws of Arithmetic'', translated and edited with an introduction by Philip A. Ebert and Marcus Rossberg. Oxford: Oxford University Press, 2013. .

Online bibliography of Frege's works and their English translations

(compiled by Edward N. Zalta, '' Stanford Encyclopedia of Philosophy''). * 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. "Ueber

philosopher
A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...

, logician
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...

, and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
...

. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy
Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United St ...

, concentrating on the philosophy of language, logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The stand ...

(1858–1932), Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...

(1872–1970), and, to some extent, Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrians, Austrian-British people, British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy o ...

(1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phi ...

, and one of the most profound philosophers of mathematics ever.
His contributions include the development of modern logic in the '' Begriffsschrift'' and work in the foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathem ...

. His book the '' Foundations of Arithmetic'' is the seminal text of the logicist
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all ...

project, and is cited by Michael Dummett
Sir Michael Anthony Eardley Dummett (27 June 1925 – 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He ...

as where to pinpoint the linguistic turn
The linguistic turn was a major development in Western philosophy during the early 20th century, the most important characteristic of which is the focusing of philosophy and the other humanities primarily on the relations between language, langua ...

. His philosophical papers "On Sense and Reference
In the philosophy of language, the distinction between sense and reference was an idea of the German philosopher and mathematician Gottlob Frege in 1892 (in his paper "On Sense and Reference"; German: "Über Sinn und Bedeutung"), reflecting the ...

" and "The Thought" are also widely cited. The former argues for two different types of meaning
Meaning most commonly refers to:
* Meaning (linguistics), meaning which is communicated through the use of language
* Meaning (philosophy), definition, elements, and types of meaning discussed in philosophy
* Meaning (non-linguistic), a general ...

and descriptivism. In ''Foundations'' and "The Thought", Frege argues for Platonism
Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at le ...

against psychologism
Psychologism is a family of philosophical positions, according to which certain psychological facts, laws, or entities play a central role in grounding or explaining certain non-psychological facts, laws, or entities. The word was coined by Johann ...

or formalism
Formalism may refer to:
* Form (disambiguation)
* Formal (disambiguation)
* Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary
* Formalism (linguistics)
* S ...

, concerning number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ca ...

s and proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

s respectively. Russell's paradox undermined the logicist project by showing Frege's Basic Law V in the ''Foundations'' to be false.
Life

Childhood (1848–69)

Frege was born in 1848 inWismar
Wismar (; Low German: ''Wismer''), officially the Hanseatic City of Wismar (''Hansestadt Wismar'') is, with around 43,000 inhabitants, the sixth-largest city of the northeastern German state of Mecklenburg-Vorpommern, and the fourth-largest city ...

, Mecklenburg-Schwerin
The Duchy of Mecklenburg-Schwerin was a duchy in northern Germany created in 1701, when Frederick William and Adolphus Frederick II divided the Duchy of Mecklenburg between Schwerin and Strelitz. Ruled by the successors of the Nikloting Hou ...

(today part of Mecklenburg-Vorpommern
Mecklenburg-Vorpommern (MV; ; nds, Mäkelborg-Vörpommern), also known by its anglicized name Mecklenburg–Western Pomerania, is a state in the north-east of Germany. Of the country's sixteen states, Mecklenburg-Vorpommern ranks 14th in po ...

). His father Carl (Karl) Alexander Frege (1809–1866) was the co-founder and headmaster of a girls' high school until his death. After Carl's death, the school was led by Frege's mother Auguste Wilhelmine Sophie Frege (née Bialloblotzky, 12 January 1815 – 14 October 1898); her mother was Auguste Amalia Maria Ballhorn, a descendant of Philipp Melanchthon
Philip Melanchthon. (born Philipp Schwartzerdt; 16 February 1497 – 19 April 1560) was a German Lutheran reformer, collaborator with Martin Luther, the first systematic theologian of the Protestant Reformation, intellectual leader of the L ...

and her father was Johann Heinrich Siegfried Bialloblotzky, a descendant of a Polish
Polish may refer to:
* Anything from or related to Poland, a country in Europe
* Polish language
* Poles, people from Poland or of Polish descent
* Polish chicken
*Polish brothers (Mark Polish and Michael Polish, born 1970), American twin screen ...

noble family who left Poland in the 17th century. Frege was a Lutheran.
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, entitled ''Hülfsbuch zum Unterrichte in der deutschen Sprache für Kinder von 9 bis 13 Jahren'' (2nd ed., Wismar 1850; 3rd ed., Wismar and Ludwigslust: Hinstorff, 1862) (Help book for teaching German to children from 9 to 13 years old), the first section of which dealt with the structure and logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

of language.
Frege studied at and graduated in 1869.Dale Jacquette, ''Frege: A Philosophical Biography'', Cambridge University Press, 2019, p. xiii. His teacher Gustav Adolf Leo Sachse (5 November 1843 – 1 September 1909), who was a poet, played the most important role in determining Frege's future scientific career, encouraging him to continue his studies at the University of Jena.
Studies at University (1869–74)

Frege matriculated at the University of Jena in the spring of 1869 as a citizen of theNorth German Confederation
The North German Confederation (german: Norddeutscher Bund) was initially a German military alliance established in August 1866 under the leadership of the Kingdom of Prussia, which was transformed in the subsequent year into a confederated st ...

. In the four semesters of his studies he attended approximately twenty courses of lectures, most of them on mathematics and physics. His most important teacher was Ernst Karl Abbe
Ernst Karl Abbe HonFRMS (23 January 1840 – 14 January 1905) was a German physicist, optical scientist, entrepreneur, and social reformer. Together with Otto Schott and Carl Zeiss, he developed numerous optical instruments. He was also a ...

(1840–1905; physicist, mathematician, and inventor). Abbe gave lectures on theory of gravity, galvanism and electrodynamics, complex analysis theory of functions of a complex variable, applications of physics, selected divisions of mechanics, and mechanics of solids. Abbe was more than a teacher to Frege: he was a trusted friend, and, as director of the optical manufacturer Carl Zeiss AG, 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 Christian Philipp Karl Snell (1806–86; subjects: use of infinitesimal analysis in geometry, analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...

of planes
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...

, analytical mechanics, optics, physical foundations of mechanics); Hermann Karl Julius Traugott Schaeffer (1824–1900; analytic geometry, applied physics, algebraic analysis, on the telegraph and other electronic machines); and the philosopher Kuno Fischer
Ernst Kuno Berthold Fischer (23 July 1824 – 5 July 1907) was a German philosopher, a historian of philosophy and a critic.
Biography
After studying philosophy at Leipzig and Halle,
became a privatdocent at Heidelberg in 1850. The Baden gov ...

(1824–1907; Kantian
Kantianism is the philosophy of Immanuel Kant, a German philosopher born in Königsberg, Prussia (now Kaliningrad, Russia). The term ''Kantianism'' or ''Kantian'' is sometimes also used to describe contemporary positions in philosophy of mind, ...

and critical philosophy).
Starting in 1871, Frege continued his studies in Göttingen, the leading university in mathematics in German-speaking territories, where he attended the lectures of Rudolf Friedrich Alfred Clebsch (1833–72; analytic geometry), Ernst Christian Julius Schering (1824–97; function theory), Wilhelm Eduard Weber (1804–91; physical studies, applied physics), Eduard Riecke (1845–1915; theory of electricity), and Hermann Lotze
Rudolf Hermann Lotze (; ; 21 May 1817 – 1 July 1881) was a German philosopher and logician. He also had a medical degree and was well versed in biology. He argued that if the physical world is governed by mechanical laws and relations, then dev ...

(1817–81; 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 or not there was a direct influence on Frege's views arising from his attending Lotze's lectures.
In 1873, Frege attained his doctorate under Ernst Christian Julius Schering, with a dissertation under the title of "Ueber 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 of projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...

's infinitely distant (imaginary) points.
Frege married Margarete Katharina Sophia Anna Lieseberg (15 February 1856 – 25 June 1904) on 14 March 1887.
Work as a logician

Though his education and early mathematical work focused primarily on geometry, Frege's work soon turned to logic. His marked a turning point in the history of logic. The ''Begriffsschrift'' broke new ground, including a rigorous treatment of the ideas of functions and variables. Frege's goal was to show that mathematics grows out oflogic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

, and in so doing, he devised techniques that separated him from the Aristotelian syllogistic but took him rather close to Stoic propositional logic.
In effect, Frege invented axiomatic
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

predicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality. Previous logic had dealt with the logical constant In logic, a logical constant of a language
Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which human
Huma ...

s ''and'', ''or'', ''if... then...'', ''not'', and ''some'' and ''all'', but iterations of these operations, especially "some" and "all", were little understood: even the distinction between a sentence like "every boy loves some girl" and "some girl is loved by every boy" could be represented only very artificially, whereas Frege's formalism had no difficulty expressing the different readings of "every boy loves some girl who loves some boy who loves some girl" and similar sentences, in complete parallel with his treatment of, say, "every boy is foolish".
A frequently noted example is that Aristotle's logic is unable to represent mathematical statements like Euclid's theorem, a fundamental statement of number theory that there are an infinite number of prime numbers. Frege's "conceptual notation", however, can represent such inferences. The analysis of logical concepts and the machinery of formalization that is essential to ''Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...

'' (3 vols., 1910–13, by Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...

, 1872–1970, and Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applica ...

, 1861–1947), to Russell's theory of descriptions
The theory of descriptions is the philosopher Bertrand Russell's most significant contribution to the philosophy of language. It is also known as Russell's theory of descriptions (commonly abbreviated as RTD). In short, Russell argued that the ...

, to Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...

's (1906–78) incompleteness theorems, and to Alfred Tarski's (1901–83) 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 larger purpose was to defend the view that arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th cent ...

is a branch of logic, a view known as logicism: unlike geometry, arithmetic was to be shown to have no basis in "intuition", and no need for non-logical axioms. Already in the 1879 ''Begriffsschrift'' important preliminary theorems, for example, a generalized form of law of trichotomy
In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.T ...

, were derived within what Frege understood to be pure logic.
This idea was formulated in non-symbolic terms in his '' The Foundations of Arithmetic'' (''Die Grundlagen der Arithmetik'', 1884). Later, in his ''Basic Laws of Arithmetic'' (''Grundgesetze der Arithmetik'', vol. 1, 1893; vol. 2, 1903; vol. 2 was published at his own expense), Frege attempted to derive, by use of his symbolism, 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 : 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 denote the extension of the predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...

''Fx'', that is, the set of all Fs, and similarly for ''Gx''. Then Basic Law V says that the predicates ''Fx'' and ''Gx'' have the same extension if and only if ∀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 that the set thus characterised ''both'' is ''and'' is not a member of itself, and is 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. Frege opened the Appendix with the exceptionally honest comment: "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion." (This letter and Frege's reply are translated in Jean van Heijenoort 1967.)
Frege's proposed remedy was subsequently shown to imply that there is but one object in the universe of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
Overview
The doma ...

, 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, for example, 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 to philosopher and mathematical logician George Boolos
George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
Life
Boolos is of Greek-Jewish descent. He graduated with an A.B. in ...

(1940–1996), who was an expert on the work of Frege. A "concept" ''F'' is "small" if the objects falling under ''F'' cannot be put into one-to-one correspondence with the universe of discourse, that is, unless: ∃''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 if second-order arithmetic
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.
A precur ...

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 ''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 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 "Frege's Logic, Theorem, and Foundations for Arithmetic".
* Frege's logic, now known as second-order logic, can be weakened to so-called predicative second-order logic. Predicative second-order logic plus Basic Law V is provably consistent by finitistic or constructive
Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...

methods, but it can interpret only very weak fragments of arithmetic.
Frege's work in logic had little international attention until 1903 when Russell wrote an appendix to '' The Principles of Mathematics'' stating his differences with Frege. The diagrammatic notation that Frege used had no antecedents (and has had no imitators since). Moreover, until Russell and Whitehead's ''Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...

'' (3 vols.) appeared in 1910–13, the dominant approach to mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

was still that of George Boole (1815–64) and his intellectual descendants, especially Ernst Schröder (1841–1902). Frege's logical ideas nevertheless spread through the writings of his student Rudolf Carnap (1891–1970) and other admirers, particularly Bertrand Russell and Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrians, Austrian-British people, British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy o ...

(1889–1951).
Philosopher

Frege is one of the founders ofanalytic philosophy
Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United St ...

, whose work on logic and language gave rise to the linguistic turn
The linguistic turn was a major development in Western philosophy during the early 20th century, the most important characteristic of which is the focusing of philosophy and the other humanities primarily on the relations between language, langua ...

in philosophy. His contributions to the philosophy of language include:
* Function and argument analysis of the proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

;
* Distinction between concept and object (''Begriff und Gegenstand'');
* Principle of compositionality
In semantics, mathematical logic and related disciplines, the principle of compositionality 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. ...

;
* Context principle In the philosophy of language, the context principle is a form of semantic holism holding that a philosopher should "never ... ask for the meaning of a word in isolation, but only in the context of a proposition" (Frege 884/1980x).
Analysis
The c ...

; and
* Distinction between the sense and reference
In the philosophy of language, the distinction between sense and reference was an idea of the German philosopher and mathematician Gottlob Frege in 1892 (in his paper "On Sense and Reference"; German: "Über Sinn und Bedeutung"), reflecting the ...

(''Sinn und Bedeutung'') of names and other expressions, sometimes said to involve a mediated reference theory A mediated reference theory (also indirect reference theory)Leszek Berezowski, ''Articles and Proper Names'', University of Wrocław, 2001, p. 67. is any semantic theory that posits that words refer to something in the external world, but insists th ...

.
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 the philosophy of language.
Sense and reference

Frege's 1892 paper, "On Sense and Reference
In the philosophy of language, the distinction between sense and reference was an idea of the German philosopher and mathematician Gottlob Frege in 1892 (in his paper "On Sense and Reference"; German: "Über Sinn und Bedeutung"), reflecting the ...

" ("Über Sinn und Bedeutung"), introduced his influential distinction between ''sense'' ("Sinn") and ''reference'' ("Bedeutung", which has also been translated as "meaning", or "denotation"). While conventional accounts of meaning took expressions to have just one feature (reference), Frege introduced the view that expressions have two different aspects of significance: their sense and their reference.
''Reference'' (or "Bedeutung") applied to proper names, where a given expression (say the expression "Tom") simply refers to the entity bearing the name (the person named Tom). Frege also held that propositions had a referential relationship with their truth-value (in other words, a statement "refers" to the truth-value it takes). 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, and there can be multiple modes of representation for the same referent.
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 Saul Kripke's famous lectures "Naming and Necessity
''Naming and Necessity'' is a 1980 book with the transcript of three lectures, given by the philosopher Saul Kripke, at Princeton University in 1970, in which he dealt with the debates of proper names in the philosophy of language. The transcrip ...

".
1924 diary

Frege's published philosophical writings were of a very technical nature and divorced from practical issues, so much so that Frege scholar Dummett expressed his "shock to discover, while reading Frege's diary, that his hero was an anti-Semite." After theGerman Revolution of 1918–19
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...

his political opinions became more radical. In the last year of his life, at the age of 76, his diary contained political opinions opposing the parliamentary system, democrats, liberals, Catholics, the French and Jews, who he thought ought to be deprived of political rights and, preferably, expelled from Germany. Frege confided "that he had once thought of himself as a liberal and was an admirer of Bismarck", but then sympathized with General Ludendorff. In an entry dated 5 May 1924 Frege expressed agreement with an article published in Houston Stewart Chamberlain's ''Deutschlands Erneuerung'' which praised Adolf Hitler
Adolf Hitler (; 20 April 188930 April 1945) was an Austrian-born German politician who was dictator of Nazi Germany, Germany from 1933 until Death of Adolf Hitler, his death in 1945. Adolf Hitler's rise to power, He rose to power as the le ...

. Frege recorded the belief that it would be best if the Jews of Germany would "get lost, or better would like to disappear from Germany." Some interpretations have been written about that time. The diary contains a critique of universal suffrage
Universal suffrage (also called universal franchise, general suffrage, and common suffrage of the common man) gives the right to vote to all adult citizens, regardless of wealth, income, gender, social status, race, ethnicity, or political stanc ...

and socialism. Frege had friendly relations with Jews in real life: among his students was Gershom Scholem
Gershom Scholem () (5 December 1897 – 21 February 1982), was a German-born Israeli philosopher and historian. Widely regarded as the founder of modern academic study of the Kaballah, Scholem was appointed the first professor of Jewish Mysticism ...

, who greatly valued his teaching, and it was he who encouraged Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrians, Austrian-British people, British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy o ...

to leave for England in order to study with Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...

. The 1924 diary was published posthumously in 1994. Frege apparently never spoke in public about his political viewpoints.
Personality

Frege was described by his students as a highly introverted person, seldom entering into dialogues with others and mostly facing the blackboard while lecturing. He was, however, known to occasionally show wit and even bitter sarcasm during his classes.Important dates

* Born 8 November 1848 inWismar
Wismar (; Low German: ''Wismer''), officially the Hanseatic City of Wismar (''Hansestadt Wismar'') is, with around 43,000 inhabitants, the sixth-largest city of the northeastern German state of Mecklenburg-Vorpommern, and the fourth-largest city ...

, Mecklenburg-Schwerin
The Duchy of Mecklenburg-Schwerin was a duchy in northern Germany created in 1701, when Frederick William and Adolphus Frederick II divided the Duchy of Mecklenburg between Schwerin and Strelitz. Ruled by the successors of the Nikloting Hou ...

.
* 1869 — attends the University of Jena.
* 1871 — attends the University of Göttingen.
* 1873 — PhD, doctor in mathematics (geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

), attained at Göttingen.
* 1874 — Habilitation at Jena; private teacher.
* 1879 — Ausserordentlicher Professor at Jena.
* 1896 — Ordentlicher Honorarprofessor at Jena.
* 1918 — retires.
* Died 26 July 1925 in Bad Kleinen
Bad Kleinen (until 1915 Kleinen) is a municipality in the Nordwestmecklenburg district, in Mecklenburg-Vorpommern, Germany. It is located on the north bank of the Schweriner See. Bad Kleinen is part of the Hamburg Metropolitan Region.
Geography ...

(now part of Mecklenburg-Vorpommern
Mecklenburg-Vorpommern (MV; ; nds, Mäkelborg-Vörpommern), also known by its anglicized name Mecklenburg–Western Pomerania, is a state in the north-east of Germany. Of the country's sixteen states, Mecklenburg-Vorpommern ranks 14th in po ...

).
Important works

Logic, foundation of arithmetic

'' Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens'' (1879), Halle an der Saale: Verlag von Louis Nebertonline version

. * In English: ''Begriffsschrift, a Formula Language, Modeled Upon That of Arithmetic, for Pure Thought'', in: J. van Heijenoort (ed.), ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931'', Harvard, MA: Harvard University Press, 1967, pp. 5–82. * In English (selected sections revised in modern formal notation): R. L. Mendelsohn, ''The Philosophy of Gottlob Frege'', Cambridge: Cambridge University Press, 2005: "Appendix A. Begriffsschrift in Modern Notation: (1) to (51)" and "Appendix B. Begriffsschrift in Modern Notation: (52) to (68)." '' Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl'' (1884), Breslau: Verlag von Wilhelm Koebner

online version

. * In English: '' The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number'', translated by J. L. Austin, Oxford: Basil Blackwell, 1950. ''Grundgesetze der Arithmetik'', Band I (1893); Band II (1903), Jena: Verlag Hermann Pohle

online version)

* In English (translation of selected sections), "Translation of Part of Frege's ''Grundgesetze der Arithmetik''," translated and edited Peter Geach and

Max Black
Max Black (24 February 1909 – 27 August 1988) was an Azerbaijani-born British-American philosopher who was a leading figure in analytic philosophy in the years after World War II. He made contributions to the philosophy of language, the philo ...

in ''Translations from the Philosophical Writings of Gottlob Frege'', New York, NY: Philosophical Library, 1952, pp. 137–158.
* In German (revised in modern formal notation): ''Grundgesetze der Arithmetik'', Korpora (portal of the University of Duisburg-Essen
The University of Duisburg-Essen (german: link=no, Universität Duisburg-Essen) is a public research university in North Rhine-Westphalia, Germany. In the 2019 ''Times Higher Education World University Rankings'', the university was awarded ...

), 2006Band I

an

Band II

. * In German (revised in modern formal notation): ''Grundgesetze der Arithmetik – Begriffsschriftlich abgeleitet. Band I und II: In moderne Formelnotation transkribiert und mit einem ausführlichen Sachregister versehen'', edited by T. Müller, B. Schröder, and R. Stuhlmann-Laeisz, Paderborn: mentis, 2009. * In English: ''Basic Laws of Arithmetic'', translated and edited with an introduction by Philip A. Ebert and Marcus Rossberg. Oxford: Oxford University Press, 2013. .

Philosophical studies

" Function and Concept" (1891) * Original: "Funktion und Begriff", anaddress
An address is a collection of information, presented in a mostly fixed format, used to give the location of a building, apartment, or other structure or a plot of land, generally using political boundaries and street names as references, along ...

to the Jenaische Gesellschaft für Medizin und Naturwissenschaft, Jena, 9 January 1891.
* In English: "Function and Concept".
"On Sense and Reference
In the philosophy of language, the distinction between sense and reference was an idea of the German philosopher and mathematician Gottlob Frege in 1892 (in his paper "On Sense and Reference"; German: "Über Sinn und Bedeutung"), reflecting the ...

" (1892)
* Original: "Über Sinn und Bedeutung", in '' Zeitschrift für Philosophie und philosophische Kritik C'' (1892): 25–50.
* In English: "On Sense and Reference", alternatively translated (in later edition) as "On Sense and Meaning".
" Concept and Object" (1892)
* Original: "Ueber Begriff und Gegenstand", in ''Vierteljahresschrift für wissenschaftliche Philosophie XVI'' (1892): 192–205.
* In English: "Concept and Object".
"What is a Function?" (1904)
* Original: "Was ist eine Funktion?", in ''Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20 February 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, the papers were published together in ''Logische Untersuchungen'', ed. G. Patzig, Vandenhoeck & Ruprecht, 1966, and English translations appeared together in ''Logical Investigations'', ed. Peter Geach, Blackwell, 1975.
* 1918–19. "Der Gedanke: Eine logische Untersuchung" ("The Thought: A Logical Inquiry"), 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. ''Jahresbericht der deutschen Mathematiker-Vereinigung XII'' (1903), 368–375. ** In English: "On the Foundations of Geometry". * 1967: ''Kleine Schriften''. (I. Angelelli, ed.). Darmstadt: Wissenschaftliche Buchgesellschaft, 1967 and Hildesheim, G. Olms, 1967. "Small Writings," a collection of most of his writings (e.g., the previous),posthumously
Posthumous may refer to:
* Posthumous award - an award, prize or medal granted after the recipient's death
* Posthumous publication – material published after the author's death
* ''Posthumous'' (album), by Warne Marsh, 1987
* ''Posthumous'' ...

published.
See also

* Frege system * List of pioneers in computer science * Neo-FregeanismNotes

References

Sources

Primary

Online bibliography of Frege's works and their English translations

(compiled by Edward N. Zalta, '' Stanford Encyclopedia of Philosophy''). * 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. "Ueber