Gottlob Frege
Frege in c. 1879
Born8 November 1848
Died26 July 1925(1925-07-26) (aged 76)
EducationUniversity of Göttingen (PhD, 1873)
University of Jena (Dr. phil. hab., 1874)
Notable workBegriffsschrift (1879)
The Foundations of Arithmetic (1884)
Basic Laws of Arithmetic (1893–1903)
Era19th-century philosophy
20th-century philosophy
RegionWestern philosophy
SchoolAnalytic philosophy
Linguistic turn
Logical objectivism
Modern Platonism[1]
Transcendental idealism[2][3] (before 1891)
Metaphysical realism[3] (after 1891)
Indirect realism[5]
Redundancy theory of truth[6]
InstitutionsUniversity of Jena
Doctoral advisorErnst Christian Julius Schering (PhD thesis advisor)
Other academic advisorsRudolf Friedrich Alfred Clebsch
Notable studentsRudolf Carnap
Main interests
Philosophy of mathematics, mathematical logic, philosophy of language
Notable ideas

Friedrich Ludwig Gottlob Frege (/ˈfrɡə/;[15] German: [ˈɡɔtloːp ˈfreːɡə]; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever.[16]

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" and "The Thought" are also widely cited. The former argues for two different types of meaning and descriptivism. In Foundations and "The Thought", Frege argues for Platonism against psychologism or formalism, concerning numbers and propositions respectively. Russell's paradox undermined the logicist project by showing Frege's Basic Law V in the Foundations to be false.


Childhood (1848–69)

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[17] and her father was Johann Heinrich Siegfried Bialloblotzky, a descendant of a Polish noble family who left Poland in the 17th century.[18] Frege was a Lutheran.[19]

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 of language.

Frege studied at Große Stadtschule Wismar [de] and graduated in 1869.[20] 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 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, analytic geometry of 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 (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; 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 (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.[20] The couple had at least two children, who unfortunately died when young. Years later they adopted a son, Alfred. Little else is known about Frege's family life, however.[21]

Work as a logician

Main article: Begriffsschrift

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 separated him from the Aristotelian syllogistic but took him rather close to Stoic propositional logic.[22]

Title page to Begriffsschrift (1879)

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.[23] 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 (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 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, 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[FxGx]. 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:

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, c. 1905

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:

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

Main article: Sense and reference

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 King of the United Kingdom", which contains the significant parts "the King of ξ" and "United Kingdom", have the same referent, namely, the person best known as King Charles III. But the sense of the word "United Kingdom" is a part of the sense of the latter expression, but no part of the sense of the "full name" of King 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".

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."[26] 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 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.[27] 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.[28] 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."[28] Some interpretations have been written about that time.[29] 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,[30][31] who greatly valued his teaching, and it was he who encouraged Ludwig Wittgenstein to leave for England in order to study with Bertrand Russell.[32] The 1924 diary was published posthumously in 1994.[33] Frege apparently never spoke in public about his political viewpoints.[citation needed]


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.[34]

Important dates

Important works

Logic, foundation of arithmetic

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

Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl (1884), Breslau: Verlag von Wilhelm Koebner (online version).

Grundgesetze der Arithmetik, Band I (1893); Band II (1903), Jena: Verlag Hermann Pohle (online version).

Philosophical studies

"Function and Concept" (1891)

"On Sense and Reference" (1892)

"Concept and Object" (1892)

"What is a Function?" (1904)

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.

Articles on geometry

See also


  1. ^ Only the proofs of Part II of the Begriffsschrift are rewritten in modern notation in this work. Partial rewriting of the proofs of Part III is included in Boolos, George, "Reading the Begriffsschrift," Mind 94(375): 331–344 (1985).
  2. ^ The journal Beiträge zur Philosophie des Deutschen Idealismus was the organ of Deutsche Philosophische Gesellschaft [de].


  1. ^ Balaguer, Mark (25 July 2016). Zalta, Edward N. (ed.). Platonism in Metaphysics. Metaphysics Research Lab, Stanford University – via Stanford Encyclopedia of Philosophy.
  2. ^ Hans Sluga, "Frege's alleged realism," Inquiry 20 (1–4):227–242 (1977).
  3. ^ a b Michael Resnik, II. Frege as Idealist and then Realist," Inquiry 22 (1–4):350–357 (1979).
  4. ^ Tom Rockmore, On Foundationalism: A Strategy for Metaphysical Realism, Rowman & Littlefield, 2004, p. 111.
  5. ^ 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).
  6. ^ a b Truth – Internet Encyclopedia of Philosophy; The Deflationary Theory of Truth (Stanford Encyclopedia of Philosophy).
  7. ^ Gottlob Frege, Grundgesetze der Arithmetik I, Jena: Verlag Hermann Pohle, 1893, §36.
  8. ^ Willard Van Orman Quine, introduction to Moses Schönfinkel's "Bausteine der mathematischen Logik", pp. 355–357, esp. 355. 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.
  9. ^ Gottlob Frege, The Foundations of Arithmetic, Northwestern University Press, 1980, p. 87.
  10. ^ Hans Sluga (1980), Gottlob Frege, Routledge, pp. 53ff.
  11. ^ 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
  12. ^ 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.
  13. ^ Tom Ricketts, Michael Potter, The Cambridge Companion to Frege, Cambridge University Press, 2010, p. 179.
  14. ^ Sundholm, B. G., "When, and why, did Frege read Bolzano?", LOGICA Yearbook 1999, 164–174 (2000).
  15. ^ "Frege". Random House Webster's Unabridged Dictionary.
  16. ^ Wehmeier, Kai F. (2006). "Frege, Gottlob". In Borchert, Donald M. (ed.). Encyclopedia of Philosophy. Vol. 3 (2 ed.). Macmillan Reference USA. ISBN 0-02-866072-2.
  17. ^ Lothar Kreiser, Gottlob Frege: Leben – Werk – Zeit, Felix Meiner Verlag, 2013, p. 11.
  18. ^ Arndt Richter, "Ahnenliste des Mathematikers Gottlob Frege, 1848–1925"
  19. ^ Frege: A Philosophical Biography. Cambridge University Press. 4 April 2019. ISBN 9780521863278.
  20. ^ a b Dale Jacquette, Frege: A Philosophical Biography, Cambridge University Press, 2019, p. xiii.
  21. ^ "Frege, Gottlob | Internet Encyclopedia of Philosophy".
  22. ^ Susanne Bobzien published in 2021 a work provocatively titled "Frege plagiarized the Stoics": Bobzien S., – In: Themes in Plato, Aristotle, and Hellenistic Philosophy, Keeling Lectures 2011–2018, p.149-206; Zalta, Ed, Frege, Stanford Encyclopedia of Philosophy
  23. ^ Horsten, Leon and Pettigrew, Richard, "Introduction" in The Continuum Companion to Philosophical Logic (Continuum International Publishing Group, 2011), p. 7.
  24. ^ Frege's Logic, Theorem, and Foundations for Arithmetic, Stanford Encyclopedia of Philosophy at
  25. ^ Burgess, John (2005). Fixing Frege. ISBN 978-0-691-12231-1.
  26. ^ Hersh, Reuben, What Is Mathematics, Really? (Oxford University Press, 1997), p. 241.
  27. ^ Michael Dummett: Frege: Philosophy of Language, p. xii.
  28. ^ a b Yvonne Sherratt (21 May 2013). Hitler's Philosophers. Yale University Press. p. 60. ISBN 978-0-300-15193-0. OCLC 1017997313.
  29. ^ 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: Forvm, vol. 36, no. 432, 20 December 1989, pp. 68–79.
  30. ^ "Frege biography".
  31. ^ "Frege, Gottlob – Internet Encyclopedia of Philosophy".
  32. ^ "Juliet Floyd, The Frege-Wittgenstein Correspondence: Interpretive Themes" (PDF). Archived (PDF) from the original on 21 May 2013.
  33. ^ 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.
  34. ^ Frege's Lectures on Logic, ed. by Erich H. Reck and Steve Awodey, Open Court Publishing, 2004, pp. 18–26.
  35. ^ Jacquette, Dale, ed. (2019), "Chronology of Major Events in Frege's Life", Frege: A Philosophical Biography, Cambridge: Cambridge University Press, pp. xiii–xiv, doi:10.1017/9781139033725.001, ISBN 978-1-139-03372-5, S2CID 242262152
  36. ^ 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.





Logic and mathematics

Historical context