**Friedrich Ludwig Gottlob Frege** (/ˈfreɪɡə/;^{[13]} German: [ˈɡɔtloːp ˈfreːɡə]; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language and mathematics. Though largely ignored during his lifetime, Giuseppe Peano (1858–1932) and Bertrand Russell (1872–1970) introduced his work to later generations of logicians and philosophers.

His contributions include the development of modern logic in the *Begriffsschrift* and work in the foundations of mathematics. His book the *Foundations of Arithmetic* is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical papers "On Sense and Reference" ("Über Sinn und Bedeutung") and "The Thought" ("Der Gedanke") are widely cited.

Frege was born in 1848 in Wismar, Mecklenburg-Schwerin (today part of Mecklenburg-Vorpommern). 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^{[14]} and her father was Johann Heinrich Siegfried Bialloblotzky, a descendant of a Polish noble family who left Poland in the 17th century.^{[15]}

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), the first section of which dealt with the structure and logic of language.

Frege studied at a *gymnasium* in Wismar and graduated in 1869. 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.

Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Confederation. 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 (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, analytical geometry of planes, analytical mechanics, optics, physical foundations of mechanics); Hermann Karl Julius Traugott Schaeffer (1824–1900; analytical geometry, applied physics, algebraic analysis, on the telegraph and other electronic machines); and the philosopher Kuno Fischer (1824–1907; Kantian 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; analytical 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 (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's infinitely distant (imaginary) points.

Frege married Margarete Katharina Sophia Anna Lieseberg (15 February 1856 – 25 June 1904) on 14 March 1887.

Though his education and early mathematical work focused primarily on geometry, Frege's work soon turned to logic. His *Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens* [*Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic*], Halle a/S: Verlag von Louis Nebert, 1879 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 of logic, and in so doing, he 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, Frege invented axiomatic 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 constants *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.^{[16]} The analysis of logical concepts and the machinery of formalization that is essential to *Principia Mathematica* (3 vols., 1910–13) (by Bertrand Russell, 1872–1970, and Alfred North Whitehead, 1861–1947), to Russell's theory of descriptions, to Kurt Gödel'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 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, were derived within what Frege understood to be pure logic.

This idea was formulated in non-symbolic terms in his *The Foundations of Arithmetic* (1884). Later, in his *Basic Laws of Arithmetic* (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 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 extension iff ∀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, 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 (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 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".^{[17]} - 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 methods, but it can interpret only very weak fragments of arithmetic.
^{[18]}

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* (3 vols.) appeared in 1910–13, the dominant approach to mathematical logic 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 (1889–1951).

Frege is one of the founders of analytic philosophy, whose work on logic and language gave rise to the linguistic turn in philosophy. His contributions to the philosophy of language include:

- Function–argument analysis of the proposition;
- Distinction between concept and object (
*Begriff und Gegenstand*); - Principle of compositionality;
- Context principle;
- Distinction between the sense and reference (
*Sinn und Bedeutung*) of names and other expressions, sometimes said to involve a mediated 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 the philosophy of language.

It should be kept in mind that Frege was employed as a mathematician, not a philosopher, and he 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 (1916-2013) and Max Black (1909–88), with the bibliographic assistance of Wittgenstein (see Geach, ed. 1975, Introduction). 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 work on logic and semantics by Polish logicians.

Frege's 1892 paper, "On Sense and Reference" ("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".

Frege's published philosophical writings were of a very technical nature and divorced from practical issues, so much so that Frege scholar Dummett expresses his "shock to discover, while reading Frege's diary, that his hero was an anti-Semite."^{[19]} After the German Revolution of 1918–19 his political opinions became more radical. In the last year of his life, at the age of 76, his diary contains extreme right-wing 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.^{[20]} Frege confided "that he had once thought of himself as a liberal and was an admirer of Bismarck", but then sympathized with General Ludendorff. Some interpretations have been written about that time.^{[21]} The diary contains a critique of universal suffrage and socialism. Frege had friendly relations with Jews in real life: among his students was Gershom Scholem,^{[22]}^{[23]} who greatly valued his teaching, and it was he who encouraged Ludwig Wittgenstein to leave for England in order to study with Bertrand Russell.^{[24]} The 1924 diary was published posthumously in 1994.^{[25]} Frege apparently never spoke in public about his political viewpoints.

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.^{[26]}

- Born 8 November 1848 in Wismar, Mecklenburg-Schwerin.
- 1869 — attends the University of Jena.
- 1871 — attends the University of Göttingen.
- 1873 — PhD, doctor in mathematics (geometry), attained at Göttingen.
- 1874 — Habilitation at Jena; private teacher.
- 1879 — Ausserordentlicher Professor at Jena.
- 1896 — Ordentlicher Honorarprofessor at Jena.
- 1917 or 1918 — retires.
- Died 26 July 1925 in Bad Kleinen (now part of Mecklenburg-Vorpommern).

*Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens* (1879), Halle a. S.

- 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. (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 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 (University of Duisburg-Essen), 2006: Band I and 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. ISBN 978-0-19-928174-9.

"Function and Concept" (1891)

- Original: "Funktion und Begriff"; in
*Jenaische Gesellschaft für Medizin und Naturwissenschaft*, Jena, 9 January 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", 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 (Internet Archive:^{[27]},^{[28]},^{[29]}); - 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*:^{[30]}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.

- 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 published.

**^**Platonism in Metaphysics (Stanford Encyclopedia of Philosophy)**^**Hans Sluga, "Frege's alleged realism,"*Inquiry*20 (1–4):227–242 (1977).- ^
^{a}^{b}Michael Resnik, "II. Frege as Idealist and then Realist,"*Inquiry*22 (1–4):350–357 (1979). **^**Tom Rockmore,*On Foundationalism: A Strategy for Metaphysical Realism*, Rowman & Littlefield, 2004, p. 111.**^**Frege criticized direct realism in his "Über Sinn und Bedeutung" (see Samuel Lebens,*Bertrand Russell and the Nature of Propositions: A History and Defence of the Multiple Relation Theory of Judgement*, Routledge, 2017, p. 34).- ^
^{a}^{b}Truth – Internet Encyclopedia of Philosophy; The Deflationary Theory of Truth (Stanford Encyclopedia of Philosophy). **^**Willard Van Orman Quine, introduction to "Bausteine der mathematischen Logik", pp. 305–316. Translated by Stefan Bauer-Mengelberg as "On the building blocks of mathematical logic" in Jean van Heijenoort (1967),*A Source Book in Mathematical Logic, 1879–1931*. Harvard University Press, pp. 355–66.**^**Hans Sluga (1980),*Gottlob Frege*, Routledge, pp. 53ff.- ^
^{a}^{b}Robert Boyce Brandom, "Frege's Technical Concepts", in*Frege Synthesized: Essays on the Philosophical and Foundational Work of G. Frege*, L. Haaparanta and J. Hintikka, Synthese Library, D. Reidel, 1986, pp. 253–295 **^**Gottfried Gabriel, "Frege, Lotze, and the Continental Roots of Early Analytic Philosophy," in: Erich H. Reck (ed.).*From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy*, Oxford University Press, 2002, pp. 39–51, esp. 44–48.**^**Tom Ricketts, Michael Potter,*The Cambridge Companion to Frege*, Cambridge University Press, 2010, p. 179.**^**Sundholm, B. G., "When, and why, did Frege read Bolzano?", LOGICA Yearbook 1999, 164–174 (2000).**^**"Frege".*Random House Webster's Unabridged Dictionary*.**^**Lothar Kreiser,*Gottlob Frege: Leben - Werk - Zeit*, Felix Meiner Verlag, 2013, p. 11.**^**Arndt Richter, "Ahnenliste des Mathematikers Gottlob Frege, 1848-1925"**^**Horsten, Leon and Pettigrew, Richard, "Introduction" in*The Continuum Companion to Philosophical Logic*(Continuum International Publishing Group, 2011), p. 7.**^**Frege's Logic, Theorem, and Foundations for Arithmetic,*Stanford Encyclopedia of Philosophy*at plato.stanford.edu**^**Burgess, John (2005).*Fixing Frege*. ISBN 978-0-691-12231-1.**^**Hersh, Reuben,*What Is Mathematics, Really?*(Oxford University Press, 1997), p. 241.**^**Michael Dummett:*Frege: Philosophy of Language*, p. xii.**^**Hans Sluga:*Heidegger's Crisis: Philosophy and Politics in Nazi Germany*, pp. 99ff. Sluga's source was an article by Eckart Menzler-Trott: "Ich wünsch die Wahrheit und nichts als die Wahrheit: Das politische Testament des deutschen Mathematikers und Logikers Gottlob Frege". In:*Forum*, vol. 36, no. 432, 20 December 1989, pp. 68–79.**^**"Frege biography".**^**"Frege, Gottlob – Internet Encyclopedia of Philosophy".**^**Juliet Floyd, The Frege-Wittgenstein Correspondence: Interpretive Themes**^**Gottfried Gabriel, Wolfgang Kienzler (editors): "Gottlob Freges politisches Tagebuch". In:*Deutsche Zeitschrift für Philosophie*, vol. 42, 1994, pp. 1057–98. Introduction by the editors on pp. 1057–66. This article has been translated into English, in:*Inquiry*, vol. 39, 1996, pp. 303–342.**^***Frege's Lectures on Logic*, ed. by Erich H. Reck and Steve Awodey, Open Court Publishing, 2004, pp. 18–26.**^***Festschrift Ludwig Boltzmann gewidmet zum sechzigsten geburtstage 20. Februar 1904. Mit einem portrait, 101 abbildungen im text und 2 tafeln*. Leipzig, J.A. Barth. 1904.**^***Festschrift Ludwig Boltzmann gewidmet zum sechzigsten geburtstage 20. Februar 1904. Mit einem portrait, 101 abbildungen im text und 2 tafeln*. Leipzig, J.A. Barth. 1904.**^***Festschrift Ludwig Boltzmann gewidmet zum sechzigsten geburtstage 20. Februar 1904. Mit einem portrait, 101 abbildungen im text und 2 tafeln*. Leipzig, J.A. Barth. 1904.**^**The journal*Beiträge zur Philosophie des Deutschen Idealismus*was the organ of Deutsche Philosophische Gesellschaft

- Online bibliography of Frege's works and their English translations (compiled by E. 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 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. Band I+II online. Partial translation of volume 1: Montgomery Furth, 1964.*The Basic Laws of Arithmetic*. Univ. of California Press. Translation of selected sections from volume 2 in Geach and Black (1980). Complete translation of both volumes: Philip A. Ebert and Marcus Rossberg, 2013,*Basic Laws of Arithmetic*. Oxford University 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). - 1918–1923. Peter Geach (editor):
*Logical Investigations*, Blackwell, 1975. - 1924. Gottfried Gabriel, Wolfgang Kienzler (editors):
*Gottlob Freges politisches Tagebuch*. In:*Deutsche Zeitschrift für Philosophie*, vol. 42, 1994, pp. 1057–98. Introduction by the editors on pp. 1057–66. This article has been translated into English, in:*Inquiry*, vol. 39, 1996, pp. 303–342. - Peter Geach and Max Black, eds., and trans., 1980.
*Translations from the Philosophical Writings of Gottlob Frege*, 3rd ed. Blackwell (1st ed. 1952).

- Philosophy

- Badiou, Alain. "On a Contemporary Usage of Frege", trans. Justin Clemens and Sam Gillespie.
*UMBR(a)*, no. 1, 2000, pp. 99–115. - 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. - Currie, Gregory, 1982.
*Frege: An Introduction to His Philosophy*. Harvester Press. - 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 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. — On Frege's theory of definitions. - Sluga, Hans, 1980.
*Gottlob Frege*. Routledge. - Nicla Vassallo, 2014,
*Frege on Thinking and Its Epistemic Significance*with Pieranna Garavaso, Lexington Books–Rowman & Littlefield, Lanham, MD, Usa. - Weiner, Joan, 1990.
*Frege in Perspective*, Cornell University Press.

- Logic and mathematics

- Anderson, D. J., and Edward Zalta, 2004, "Frege, Boolos, and Logical Objects,"
*Journal of Philosophical Logic 33*: 1–26. - Blanchette, Patricia, 2012,
*Frege's Conception of Logic*. Oxford: Oxford University Press, 2012 - 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 on Frege's theorem and the logicist approach to the foundation of arithmetic. - Dummett, Michael, 1991.
*Frege: Philosophy of Mathematics*. Harvard University Press. - Demopoulos, William, ed., 1995.
*Frege's Philosophy of Mathematics*. Harvard Univ. Press. — Papers exploring Frege'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, Donald A., 1982.
*Frege, Dedekind, and Peano on the foundations of arithmetic*. Methodology and Science Foundation, 2. Van Gorcum & Co., Assen, 1982. - Gillies, Donald: The Fregean revolution in logic. Revolutions in mathematics, 265–305, Oxford Sci. Publ., Oxford Univ. Press, New York, 1992.
- Irvine, Andrew David, 2010, "Frege on Number Properties,"
*Studia Logica,*96(2): 239-60. - Charles Parsons, 1965, "Frege's Theory of Number." Reprinted with Postscript in Demopoulos (1965): 182–210. The starting point of the ongoing sympathetic reexamination of Frege's logicism.
- Gillies, Donald: The Fregean revolution in logic. Revolutions in mathematics, 265–305, Oxford Sci. Publ., Oxford Univ. Press, New York, 1992.
- Heck, Richard G., Jr:
*Frege's Theorem*. Oxford: Oxford University Press, 2011 - Heck, Richard G., Jr:
*Reading Frege's Grundgesetze*. Oxford: Oxford University Press, 2013 - 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.

- Historical context

- Everdell, William R. (1997),
*The First Moderns: Profiles in the Origins of Twentieth Century Thought*, Chicago: University of Chicago Press

- Works by or about Gottlob Frege at Internet Archive
- Frege at Genealogy Project
- A comprehensive guide to Fregean material available on the web by Brian Carver.
- Frege, Gottlob – Internet Encyclopedia of Philosophy
- Stanford Encyclopedia of Philosophy:
- Internet Encyclopedia of Philosophy:
- Gottlob Frege — by Kevin C. Klement.
- Frege and Language — by Dorothea Lotter.

- Metaphysics Research Lab: Gottlob Frege.
- Frege on Being, Existence and Truth.
- O'Connor, John J.; Robertson, Edmund F., "Gottlob Frege",
*MacTutor History of Mathematics archive*, University of St Andrews. - Begriff, a LaTeX package for typesetting Frege's logic notation, earlier version
- grundgesetze, a LaTeX package for typesetting Frege's logic notation, mature version
- Frege's Basic Laws of Arithmetic, info website, incl. corrigenda and LaTeX typesetting tool — by P.A. Ebert and M. Rossberg

1925 in philosophy

BegriffsschriftBegriffsschrift (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.

Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as "a formula language, modeled on that of arithmetic, of pure thought." Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator (despite that, in the foreword Frege clearly denies that he achieved this aim, and also that his main aim would be constructing an ideal language like Leibniz's, which Frege declares to be a quite hard and idealistic, however not impossible, task). Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter century.

Concept and objectIn the philosophy of language, the distinction between concept and object is attributable to the German philosopher Gottlob Frege.

Context principleIn 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 [1884/1980] x).

Cora DiamondCora Diamond (born 1937) is an American philosopher who works on Ludwig Wittgenstein, Gottlob Frege, moral philosophy, political philosophy, philosophy of language, and philosophy and literature. She is currently the Kenan Professor of Philosophy Emerita at the University of Virginia.

David Bell (philosopher)David Andrew Bell (born 1947) is a British philosopher. He is emeritus professor of philosophy at the University of Sheffield, He studied in Dublin (Trinity College), Göttingen (Georg-August Universität) and Canada (McMaster University), and is best known for his work on the philosophers Gottlob Frege, Immanuel Kant, and Edmund Husserl, and also on topics such as solipsism, phenomenology, the theory of thought and judgement, and the history of the Analytic Tradition.

Bell's awards include: Radcliffe Research Fellowship (1986–87); Alexander von Humboldt Research Fellowship (1988); British Academy Research Readership (1993); and the Alexander von Humboldt Prize in Philosophy (1995). He has held the posts of Visiting Professor, Institute of Philosophy, University of Leuven, (1987); Honorary Professor, University of Keele (1993–96); and Visiting Professor, Ludwig Maximillians Universität München (1994) He was a Fellow of the Wissenschaftskolleg zu Berlin [Institute of Advanced Study] (1995–96). And in 2001-2 he was President of the Mind Association.Bell has been influential in seeking to integrate the Analytic (predominantly anglophone) and the Continental (mainly French and German) traditions in philosophy. In 1993, together with Mark Sacks, he was instrumental in founding a new journal, The European Journal of Philosophy with the aim of proving 'a platform to which those both inside and outside Europe can turn to find some of the diversity ... in European philosophy', and thus overcome the insularity and at times hostility that has characterized aspects of that philosophy during the last century. In 2015 the EJP was voted among the top 20 philosophy journals world-wide. In 1999 Bell published an influential study of Husserl, in which analytic techniques were applied to a central figure of continental phenomenology.

Direct reference theoryA direct reference theory (also called referentialism or referential realism) is a theory of language that claims that the meaning of a word or expression lies in what it points out in the world. The object denoted by a word is called its referent. Criticisms of this position are often associated with Ludwig Wittgenstein.In the 19th century, mathematician and philosopher Gottlob Frege argued against it, and contrasted it with mediated reference theory. In 1953, with his Philosophical Investigations, Wittgenstein argued against referentialism, famously saying that "the meaning of a word is its use." Direct reference theory is a position typically associated with logical positivism and analytical philosophy. Logical positivist philosophers in particular have significantly devoted their efforts in countering positions of the like of Wittgenstein's, and they aim at creating a "perfectly descriptive language" purified from ambiguities and confusions.

Frege's puzzlesFrege's puzzles are puzzles about the semantics of proper names, although related puzzles also arise in the case of indexicals. Gottlob Frege (1848–1925) introduced the puzzle at the beginning of his article "Über Sinn und Bedeutung" ("On Sense and Reference") in 1892 in one of the most influential articles in analytic philosophy and philosophy of language.

Frege's theoremIn metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle. It was first proven, informally, by Gottlob Frege in his Die Grundlagen der Arithmetik (The Foundations of Arithmetic), published in 1884, and proven more formally in his Grundgesetze der Arithmetik (Basic Laws of Arithmetic), published in two volumes, in 1893 and 1903. The theorem was re-discovered by Crispin Wright in the early 1980s and has since been the focus of significant work. It is at the core of the philosophy of mathematics known as neo-logicism (at least of the Scottish School variety).

Frege–Church ontologyThe Frege–Church ontology is an ontology, a theory of existence. Everything is considered as being in three categories, object (referent, denotation), name, or concept (sense). The ontology was developed by Alonzo Church based on ideas of Gottlob Frege to resolve some paradoxes. The ontology is related to certain modal logics.

Function and Concept"Function and Concept" (German: Über Funktion und Begriff, "On Function and Concept") is an article by Gottlob Frege, published in 1891. The article involves a clarification of his earlier distinction between concepts and objects.

History of Early Analytic Philosophy SocietyThe History of Early Analytic Philosophy Society (HEAPS) is a philosophical society founded to study early analytic philosophy. The society examines the work of Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein, Frank P. Ramsey, G. E. Moore,and other early contributors to the field. The Society sponsors conference sessions in conjunction with the American Philosophical Association.

Logical InvestigationsLogical Investigations (German: Logische Untersuchungen) can refer to:

Logical Investigations, 1840 work by Friedrich Adolf Trendelenburg

Logical Investigations, 1900–1901 work by Edmund Husserl

Logical Investigations, 1918–1923 work by Gottlob Frege

Mediated reference theoryA mediated reference theory (also indirect reference theory) is any 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. It thus stands opposed to the theory of direct reference. Gottlob Frege is a well-known advocate of mediated reference theories. Similar theories were widely held in the middle of the twentieth century by philosophers such as Peter Strawson and John Searle.

Mediated reference theories are contrasted with theories of direct reference.

Saul Kripke, a proponent of direct reference theory, in his Naming and Necessity dubbed mediated reference theory the Frege–Russell view and criticized it. Subsequent scholarship refuted the claim that Bertrand Russell's views on reference theory were the same as Frege's, since Russell was also a proponent of direct reference theory.

Principle of compositionalityIn mathematics, semantics, and philosophy of language, 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. This principle is also called Frege's principle, because Gottlob Frege is widely credited for the first modern formulation of it. However, the idea appears already among Indian philosophers of grammar such as Yāska, and also in Plato's work such as in Theaetetus.

Besides, the principle was never explicitly stated by Frege, and it was arguably already assumed by George Boole decades before Frege's work.

The Foundations of ArithmeticThe Foundations of Arithmetic (German: Die Grundlagen der Arithmetik) is a book by Gottlob Frege, published in 1884, which investigates the philosophical foundations of arithmetic. Frege refutes other theories of number and develops his own theory of numbers. The Grundlagen also helped to motivate Frege's later works in logicism. The book was not well received and was not read widely when it was published. It did, however, draw the attentions of Bertrand Russell and Ludwig Wittgenstein, who were both heavily influenced by Frege's philosophy. An English translation was published (Oxford, 1950) by J. L. Austin, with a second edition in 1960.

Turnstile (symbol)In mathematical logic and computer science the symbol has taken the name **turnstile** because of its resemblance to a typical turnstile if viewed from above. It is also referred to as **tee** and is often read as "yields", "proves", "satisfies" or "entails". The symbol was first used by Gottlob Frege in his 1879 book on logic, *Begriffsschrift*.

In TeX, the turnstile symbol is obtained from the command `\vdash`. In Unicode, the turnstile symbol (⊢) is called **right tack** and is at code point U+22A2. (Code point U+22A6 is named *assertion sign* (⊦).) On a typewriter, a turnstile can be composed from a vertical bar (|) and a dash (–). In LaTeX there is a turnstile package which issues this sign in many ways, and is capable of putting labels below or above it, in the correct places.

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