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Gottlob Frege
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Life
Childhood (1848â€“69)
Frege was born in 1848 in Wismar, Mecklenburg-Schwerin (today part of Mecklenburg-Vorpommern). His father Carl (Karl) Alexander Frege (1809â€“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 (nÃ©e Bialloblotzky, 12 January 1815 â€“ 14 October 1898); her mother was Auguste Amalia Maria Ballhorn, a descendant of Philipp MelanchthonLothar Kreiser, Gottlob Frege: Leben - Werk - Zeit, Felix Meiner Verlag, 2013, p. 11. and her father was Johann Heinrich Siegfried Bialloblotzky, a descendant of a Polish noble family who left Poland in the 17th century.Arndt Richter, "Ahnenliste des Mathematikers Gottlob Frege, 1848-1925"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 9â€“13, entitled HÃ¼lfsbuch zum Unterrichte in der deutschen Sprache fÃ¼r 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 â€“ 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: Jena and GÃ¶ttingen (1869â€“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 (1840â€“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 (1806â€“86; subjects: use of infinitesimal analysis in geometry, analytical geometry of planes, analytical mechanics, optics, physical foundations of mechanics); Hermann Karl Julius Traugott Schaeffer (1824â€“1900; analytical geometry, applied physics, algebraic analysis, on the telegraph and other electronic machines); and the philosopher Kuno Fischer (1824â€“1907; Kantian and critical philosophy).Starting in 1871, Frege continued his studies in GÃ¶ttingen, the leading university in mathematics in German-speaking territories, where he attended the lectures of Rudolf Friedrich Alfred Clebsch (1833â€“72; analytical geometry), Ernst Christian Julius Schering (1824â€“97; function theory), Wilhelm Eduard Weber (1804â€“91; physical studies, applied physics), Eduard Riecke (1845â€“1915; theory of electricity), and Hermann Lotze (1817â€“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 imaginÃ¤ren 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 â€“ 25 June 1904) on 14 March 1887.Work as a logician
Though his education and early mathematical work focused primarily on geometry, Frege's work soon turned to logic. His {{Citation | title = Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens | place = Halle a/S | publisher = Verlag von Louis Nebert | year = 1879 |trans-title=Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic}} 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.(File:Begriffsschrift Titel.png|right|thumb|250px|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.Horsten, Leon and Pettigrew, Richard, "Introduction" in The Continuum Companion to Philosophical Logic (Continuum International Publishing Group, 2011), p. 7. The analysis of logical concepts and the machinery of formalization that is essential to Principia Mathematica (3 vols., 1910â€“13) (by Bertrand Russell, 1872â€“1970, and Alfred North Whitehead, 1861â€“1947), to Russell's theory of descriptions, to Kurt GÃ¶del's (1906â€“78) incompleteness theorems, and to Alfred Tarski's (1901â€“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 {{nowrap|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 âˆ€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 âˆ€x[Fx â†” 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 (1940â€“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: âˆƒR[R is 1-to-1 & âˆ€xâˆƒy(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 âˆ€x(Fx â†” 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 Fs is the same as the number of Gs if and only if the Fs can be put into a one-to-one correspondence with the Gs. This principle, too, is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic. This result is termed Frege's theorem because it was noticed that in developing arithmetic, Frege's use of Basic Law V is restricted to a proof of Hume's principle; it is from this, in turn, that arithmetical principles are derived. On Hume's principle and Frege's theorem, see "Frege's Logic, Theorem, and Foundations for Arithmetic".Frege's Logic, Theorem, and Foundations for Arithmetic, Stanford Encyclopedia of Philosophy at plato.stanford.edu
- 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.BOOK, Burgess, John, Fixing Frege, 2005, 0-691-12231-8,
Philosopher
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:- Functionâ€“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.
Sense and reference
Frege's 1892 paper, "On Sense and Reference" ("Ãœber 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 Î¾" 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".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 expresses his "shock to discover, while reading Frege's diary, that his hero was an anti-Semite."Hersh, Reuben, What Is Mathematics, Really? (Oxford University Press, 1997), p. 241. After the German Revolution of 1918â€“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.Michael Dummett: Frege: Philosophy of Language, p. xii. Frege confided "that he had once thought of himself as a liberal and was an admirer of Bismarck", but then sympathized with General Ludendorff and Adolf Hitler. Some interpretations have been written about that time.Hans Sluga: Heidegger's Crisis: Philosophy and Politics in Nazi Germany, pp. 99ff. Sluga's source was an article by Eckart Menzler-Trott: "Ich wÃ¼nsch 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. 68â€“79. 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 who much valued his teacher;WEB,weblink Frege biography, WEB,weblink Frege, Gottlob â€“ Internet Encyclopedia of Philosophy, and he encouraged Ludwig Wittgenstein to leave for England.Juliet Floyd, The Frege-Wittgenstein Correspondence: Interpretive Themes The 1924 diary was published posthumously in 1994.Gottfried Gabriel, Wolfgang Kienzler (editors): "Gottlob Freges politisches Tagebuch". In: Deutsche Zeitschrift fÃ¼r Philosophie, vol. 42, 1994, pp. 1057â€“98. Introduction by the editors on pp. 1057â€“66. This article has been translated into English, in: Inquiry, vol. 39, 1996, pp. 303â€“342. Frege apparently never spoke in public about his political viewpoints.Personality
Frege was described by his students as a highly introverted person, seldom entering into dialogue, mostly facing the blackboard while lecturing though being witty and sometimes bitterly sarcastic.Frege's Lectures on Logic, ed. by Erich H. Reck and Steve Awodey, Open Court Publishing, 2004, pp. 18â€“26.Important dates
- Born 8 November 1848 in Wismar, Mecklenburg-Schwerin.
- 1869 â€” attends the University of Jena.
- 1871 â€” attends the University of GÃ¶ttingen.
- 1873 â€” PhD, doctor in mathematics (geometry), attained at GÃ¶ttingen.
- 1874 â€” Habilitation at Jena; private teacher.
- 1879 â€” Ausserordentlicher Professor at Jena.
- 1896 â€” Ordentlicher Honorarprofessor at Jena.
- 1917 or 1918 â€” retires.
- Died 26 July 1925 in Bad Kleinen (now part of Mecklenburg-Vorpommern).
Important works
Logic, foundation of arithmetic
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 GÃ¶del: A Source Book in Mathematical Logic, 1879-1931, Harvard, MA: Harvard University Press, 1967, pp. 5â€“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)."
- In English: The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, translated by J. L. Austin, Oxford: Basil Blackwell, 1950.
- 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. 137â€“158.
- In German (revised in modern formal notation): Grundgesetze der Arithmetik â€“ Begriffsschriftlich abgeleitet. Band I und II: In moderne Formelnotation transkribiert und mit einem ausfÃ¼hrlichen Sachregister versehen, edited by T. MÃ¼ller, B. SchrÃ¶der, 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}}.
Philosophical studies
"Function and Concept" (1891)- Original: "Funktion und Begriff"; in Jenaische Gesellschaft fÃ¼r Medizin und Naturwissenschaft, Jena, 9 January 1891;
- In English: "Function and Concept''.
- Original: "Ãœber Sinn und Bedeutung", in Zeitschrift fÃ¼r Philosophie und philosophische Kritik C (1892): 25â€“50;
- In English: "On Sense and Reference", alternatively translated (in later edition) as "On Sense and Meaning".
- Original: "Ueber Begriff und Gegenstand", in Vierteljahresschrift fÃ¼r wissenschaftliche Philosophie XVI (1892): 192â€“205;
- In English: "Concept and Object".
- Original: "Was ist eine Funktion?", in Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20 February 1904, S. Meyer (ed.), Leipzig, 1904, pp. 656â€“666 (Internet Archive: weblink, weblink, weblink);
- In English: "What is a Function?".
- 1918â€“19. "Der Gedanke: Eine logische Untersuchung" ("The Thought: A Logical Inquiry"), in BeitrÃ¤ge zur Philosophie des Deutschen Idealismus I:The journal BeitrÃ¤ge zur Philosophie des Deutschen Idealismus was the organ of {{Interlanguage link multi|Deutsche Philosophische Gesellschaft|de}}. 58â€“77.
- 1918â€“19. "Die Verneinung" ("Negation") in BeitrÃ¤ge zur Philosophie des Deutschen Idealismus I: 143â€“157.
- 1923. "GedankengefÃ¼ge" ("Compound Thought"), in BeitrÃ¤ge zur Philosophie des Deutschen Idealismus III: 36â€“51.
Articles on geometry
- 1903: "Ãœber die Grundlagen der Geometrie". II. Jahresbericht der deutschen Mathematiker-Vereinigung XII (1903), 368â€“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.
See also
{{-}}References
{{Reflist}}Sources
Primary
- 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 GÃ¶del: A Source Book in Mathematical Logic, 1879â€“1931. Harvard University Press.
- 1884. Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung Ã¼ber 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. "Ãœber Sinn und Bedeutung" in Zeitschrift fÃ¼r Philosophie und philosophische Kritik 100:25â€“50. Translation: "On Sense and Reference" in Geach and Black (1980).
- 1892b. "Ueber Begriff und Gegenstand" in Vierteljahresschrift fÃ¼r wissenschaftliche Philosophie 16:192â€“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: 656â€“666. Translation: "What is a Function?" in Geach and Black (1980).
- 1918â€“1923. Peter Geach (editor): Logical Investigations, Blackwell, 1975.
- 1924. Gottfried Gabriel, Wolfgang Kienzler (editors): Gottlob Freges politisches Tagebuch. In: Deutsche Zeitschrift fÃ¼r Philosophie, vol. 42, 1994, pp. 1057â€“98. Introduction by the editors on pp. 1057â€“66. This article has been translated into English, in: Inquiry, vol. 39, 1996, pp. 303â€“342.
- Peter Geach and Max Black, eds., and trans., 1980. Translations from the Philosophical Writings of Gottlob Frege, 3rd ed. Blackwell (1st ed. 1952).
Secondary
- Philosophy
- Badiou, Alain. "On a Contemporary Usage of Frege", trans. Justin Clemens and Sam Gillespie. UMBR(a), no. 1, 2000, pp. 99â€“115.
- Baker, Gordon, and P.M.S. Hacker, 1984. Frege: Logical Excavations. Oxford University Press. â€” 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. â€” On the Frege-Husserl-Cantor triangle.
- Kenny, Anthony, 1995. Frege â€” An introduction to the founder of modern analytic philosophy. Penguin Books. â€” Excellent non-technical introduction and overview of Frege's philosophy.
- Klemke, E.D., ed., 1968. Essays on Frege. University of Illinois Press. â€” 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. â€” 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 Booksâ€“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: 1â€“26.
- Blanchette, Patricia, 2012, Frege's Conception of Logic. Oxford: Oxford University Press, 2012
- Burgess, John, 2005. Fixing Frege. Princeton Univ. Press. â€” A critical survey of the ongoing rehabilitation of Frege's logicism.
- Boolos, George, 1998. Logic, Logic, and Logic. MIT Press. â€” 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. â€” 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: 301â€“11.
- Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870â€“1940. Princeton University Press. â€” 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, 265â€“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): 182â€“210. The starting point of the ongoing sympathetic reexamination of Frege's logicism.
- Gillies, Donald: The Fregean revolution in logic. Revolutions in mathematics, 265â€“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. â€” A systematic exposition and a scope-restricted defense of Frege's Grundlagen conception of numbers.
- Historical context
- {{Citation|last=Everdell|first=William R. |year=1997 |title=The First Moderns: Profiles in the Origins of Twentieth Century Thought |location= Chicago |publisher=University of Chicago Press}}
External links
{{Sister project links|wikt=no|commons=Category:Gottlob Frege|b=no|n=no|q=Gottlob Frege|s=Author:Gottlob Frege|v=no|species=no }}- {{Internet Archive author |sname=Gottlob Frege}}
- Frege at Genealogy Project
- A comprehensive guide to Fregean material available on the web by Brian Carver.
- Stanford Encyclopedia of Philosophy:
- "Gottlob Frege" â€” by Edward Zalta.
- "Frege's Logic, Theorem, and Foundations for Arithmetic" â€” by Edward Zalta.
- Internet Encyclopedia of Philosophy:
- Gottlob Frege â€” by Kevin C. Klement.
- Frege and Language â€” by Dorothea Lotter.
- Metaphysics Research Lab: Gottlob Frege.
- Frege on Being, Existence and Truth.
- {{MacTutor Biography|id=Frege}}
- 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 â€” by P.A. Ebert and M. Rossberg
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