Friedrich Ludwig Gottlob Frege (/ˈfreɪɡə/;[7]German:[ˈɡɔtloːpˈfreːɡə]; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at theUniversity of Jena, and is understood by many to be the father ofanalytic philosophy, concentrating on thephilosophy of language,logic, andmathematics. 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 sinceAristotle, and one of the most profound philosophers of mathematics ever.[8]
Frege was born in 1848 inWismar,Mecklenburg-Schwerin (today part ofMecklenburg-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 ofPhilipp Melanchthon[9] and her father was Johann Heinrich Siegfried Bialloblotzky, a descendant of aPolish noble family who left Poland in the 17th century.[10] Frege was a Lutheran.[11]
In childhood, Frege encountered philosophies that would guide his future scientific career. For example, his father wrote atextbook on the German language for children aged 9–13, entitledHülfsbuch 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 andlogic oflanguage.
Frege studied atGroße Stadtschule Wismar [de] and graduated in 1869.[12] Teacher of mathematics and natural science Gustav Adolf Leo Sachse (1843–1909), who was also a poet, played an important role in determining Frege's future scientific career, encouraging him to continue his studies at his ownalma mater theUniversity of Jena.[13]
Frege matriculated at the University of Jena in the spring of 1869 as a citizen of theNorth 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 wasErnst 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.[citation needed]
Starting in 1871, Frege continued his studies in Göttingen, the leading university in mathematics in German-speaking territories, where he attended the lectures ofRudolf 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), andHermann 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.[citation needed]
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 ofprojective geometry's infinitely distant (imaginary) points.[citation needed]
Frege married Margarete Katharina Sophia Anna Lieseberg (15 February 1856 – 25 June 1904) on 14 March 1887.[12] 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.[14]
Though his education and early mathematical work focused primarily on geometry, Frege's work soon turned to logic. HisBegriffsschrift, 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. TheBegriffsschrift broke new ground, including a rigorous treatment of the ideas offunctions andvariables. Frege's goal was to show that mathematics grows out oflogic, and in so doing, he devised techniques that separated him from the Aristotelian syllogistic but took him rather close to Stoic propositional logic.[15]
Title page toBegriffsschrift (1879)
In effect, Frege inventedaxiomaticpredicate logic, in large part thanks to his invention ofquantified variables, which eventually became ubiquitous inmathematics and logic, and which solved theproblem of multiple generality. Previous logic had dealt with thelogical constantsand,or,if... then...,not, andsome andall, 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 likeEuclid's theorem, a fundamental statement of number theory that there are an infinite number ofprime numbers. Frege's "conceptual notation", however, can represent such inferences.[16] The analysis of logical concepts and the machinery of formalization that is essential toPrincipia Mathematica (3 vols., 1910–13, byBertrand Russell, 1872–1970, andAlfred North Whitehead, 1861–1947), to Russell'stheory of descriptions, toKurt Gödel's (1906–78)incompleteness theorems, and toAlfred 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 thatarithmetic is a branch of logic, a view known aslogicism: unlike geometry, arithmetic was to be shown to have no basis in "intuition", and no need for non-logical axioms. Already in the 1879Begriffsschrift important preliminary theorems, for example, a generalized form oflaw of trichotomy, were derived within what Frege understood to be pure logic.
This idea was formulated in non-symbolic terms in hisThe Foundations of Arithmetic (Die Grundlagen der Arithmetik, 1884). Later, in hisBasic 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 hisBegriffsschrift, though not without some significant changes. The one truly new principle was one he called theBasic Law V: the "value-range" of the functionf(x) is the same as the "value-range" of the functiong(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 theextension of thepredicateFx, that is, the set of all Fs, and similarly forGx. Then Basic Law V says that the predicatesFx andGx have the same extensionif 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 theGrundgesetze was about to go to press in 1903, showing thatRussell's paradox could be derived from Frege's Basic Law V. It is easy to define the relation ofmembership of a set or extension in Frege's system; Russell then drew attention to "the set of thingsx that are such thatx is not a member ofx". The system of theGrundgesetze entails that the set thus characterisedboth isand 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 inJean van Heijenoort 1967.)
Frege's proposed remedy was subsequently shown to imply that there is but one object in theuniverse 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 theGrundgesetze 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 logicianGeorge Boolos (1940–1996), who was an expert on the work of Frege. A "concept"F is "small" if the objects falling underF 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 neitherF norG is small or ∀x(Fx ↔Gx). V* is consistent ifsecond-order arithmetic is, and suffices to prove the axioms of second-order arithmetic.
Basic Law V can simply be replaced withHume's principle, which says that the number ofFs is the same as the number ofGs if and only if theFs can be put into a one-to-one correspondence with theGs. This principle, too, is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic. This result is termedFrege's theorem because it was noticed that in developing arithmetic, Frege's use of Basic Law V is restricted to a proof of Hume's principle; it is from this, in turn, that arithmetical principles are derived. On Hume's principle and Frege's theorem, see "Frege's Logic, Theorem, and Foundations for Arithmetic".[17]
Frege's logic, now known assecond-order logic, can be weakened to so-calledpredicative second-order logic. Predicative second-order logic plus Basic Law V is provably consistent byfinitistic orconstructive 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 toThe 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'sPrincipia Mathematica (3 vols.) appeared in 1910–13, the dominant approach tomathematical logic was still that ofGeorge Boole (1815–64) and his intellectual descendants, especiallyErnst Schröder (1841–1902). Frege's logical ideas nevertheless spread through the writings of his studentRudolf Carnap (1891–1970) and other admirers, particularly Bertrand Russell andLudwig Wittgenstein (1889–1951).[citation needed]
As a philosopher of mathematics, Frege attacked thepsychologistic 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.
Frege's 1892 paper, "On Sense and Reference" ("Über Sinn und Bedeutung"), introduced his influential distinction betweensense ("Sinn") andreference ("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 toproper 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, thesense (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 unanalysable 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 samereferent, namely, the person best known asKing Charles III. But thesense 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 bySaul 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 scholarDummett expressed his "shock to discover, while reading Frege's diary, that his hero was an anti-Semite."[19] After theGerman 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.[20] Frege confided "that he had once thought of himself as a liberal and was an admirer ofBismarck", but then sympathized with GeneralLudendorff. In an entry dated 5 May 1924 Frege expressed agreement with an article published inHouston Stewart Chamberlain'sDeutschlands Erneuerung which praisedAdolf Hitler.[21] 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."[21] Some interpretations have been written about that time.[22] The diary contains a critique ofuniversal suffrage and socialism. Frege had friendly relations with Jews in real life: among his students wasGershom Scholem,[23][24] who greatly valued his teaching, and it was he who encouragedLudwig Wittgenstein to leave for England in order to study withBertrand Russell.[25] The 1924 diary was published posthumously in 1994.[26]
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.[27]
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)."[a]
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'sGrundgesetze der Arithmetik," translated and editedPeter Geach andMax Black inTranslations 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 – 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.ISBN978-0-19-928174-9.
Original: "Ueber Begriff und Gegenstand", inVierteljahresschrift für wissenschaftliche Philosophie XVI (1892): 192–205.
In English: "Concept and Object".
"What is a Function?" (1904)
Original: "Was ist eine Funktion?", inFestschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20 February 1904, S. Meyer (ed.), Leipzig, 1904, pp. 656–666.[29]
In English: "What is a Function?".
Logical Investigations (1918–1923). Frege intended that the following three papers be published together in a book titledLogische Untersuchungen (Logical Investigations). Though the German book never appeared, the papers were published together inLogische Untersuchungen, ed. G. Patzig, Vandenhoeck & Ruprecht, 1966, and English translations appeared together inLogical Investigations, ed. Peter Geach, Blackwell, 1975.
1918–19. "Der Gedanke: Eine logische Untersuchung" ("The Thought: A Logical Inquiry"), inBeiträge zur Philosophie des Deutschen Idealismus I:[b] 58–77.
1918–19. "Die Verneinung" ("Negation") inBeiträge zur Philosophie des Deutschen Idealismus I: 143–157.
1923. "Gedankengefüge" ("Compound Thought"), inBeiträge 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.
^Only the proofs of Part II of theBegriffsschrift are rewritten in modern notation in this work. Partial rewriting of the proofs of Part III is included inBoolos, George, "Reading theBegriffsschrift,"Mind94(375): 331–344 (1985).
^Balaguer, Mark (25 July 2016). Zalta, Edward N. (ed.).Platonism in Metaphysics. Metaphysics Research Lab, Stanford University – via Stanford Encyclopedia of Philosophy.
^Tom Rockmore,On Foundationalism: A Strategy for Metaphysical Realism, Rowman & Littlefield, 2004, p. 111.
^Frege criticizeddirect 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).
^Horsten, Leon and Pettigrew, Richard, "Introduction" inThe Continuum Companion to Philosophical Logic (Continuum International Publishing Group, 2011), p. 7.
^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.http://forvm.contextxxi.org/-no-432-.html
^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 andSteve Awodey, Open Court Publishing, 2004, pp. 18–26.
1879.Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: Louis Nebert. Translation:Concept Script, a formal language of pure thought modelled upon that of arithmetic, by S. Bauer-Mengelberg inJean Van Heijenoort, ed., 1967.From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press.
1884.Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. Translation:J. L. Austin, 1974.The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, 2nd ed. Blackwell.
1891. "Funktion und Begriff." Translation: "Function and Concept" in Geach and Black (1980).
1892a. "Über Sinn und Bedeutung" inZeitschrift für Philosophie und philosophische Kritik 100:25–50. Translation: "On Sense and Reference" in Geach and Black (1980).
1892b. "Ueber Begriff und Gegenstand" inVierteljahresschrift 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 onlineArchived 17 June 2022 at theWayback Machine. 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 andMax Black, eds., and trans., 1980.Translations from the Philosophical Writings of Gottlob Frege, 3rd ed. Blackwell (1st ed. 1952).
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.
------, 1981.The Interpretation of Frege's Philosophy. Harvard University Press.
Hill, Claire Ortiz, 1991.Word and Object in Husserl, Frege and Russell: The Roots of Twentieth-Century Philosophy. Athens OH: Ohio University Press.
------, and Rosado Haddock, G. E., 2000.Husserl or Frege: Meaning, Objectivity, and Mathematics. Open Court. — On the Frege-Husserl-Cantor triangle.
Kenny, Anthony, 1995.Frege – An introduction to the founder of modern analytic philosophy. Penguin Books. — Excellent non-technical introduction and overview of Frege's philosophy.
Klemke, E.D., ed., 1968.Essays on Frege. University of Illinois Press. — 31 essays by philosophers, grouped under three headings: 1.Ontology; 2.Semantics; and 3.Logic andPhilosophy of Mathematics.
Ferreira, F. andWehmeier, K., 2002, "On the consistency of the Delta-1-1-CA fragment of Frege'sGrundgesetze,"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 Kimberly:Frege's Theorem. Oxford: Oxford University Press, 2011
Wright, Crispin, 1983.Frege's Conception of Numbers as Objects. Aberdeen University Press. — A systematic exposition and a scope-restricted defense of Frege'sGrundlagen conception of numbers.
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