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# Gottlob Frege

Friedrich Ludwig Gottlob Frege (8 November 1848 – 26 July 1925) was a German mathematician who became a logician and philosopher. He was one of the founders of modern logic, and made major contributions to the foundations of mathematics. As a philosopher, he is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and mathematics. Although he was mainly ignored by the intellectual world when he published his writings, it was Giuseppe Peano (1858–1932) and later Bertrand Russell (1872–1970) who helped introduce his work to later generations of logicians and philosophers.

Life

Childhood (1848–1869)

Frege was born in 1848 in Wismar, in the state of Mecklenburg-Schwerin ( the modern German federal state Mecklenburg-Vorpommern ). His father Carl ( Karl ) Alexander Frege ( 3 Aug 1809 - 30 Nov 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 Jan 1815 - 14 Oct 1898 ), who apparently was of Polish extraction.

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, titled 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 Nov 1843 - 1 Sept 1909 ) ( also 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 Göttingen (1869 – 1874)

Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Federation. 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, 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 (1806–1886) (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 famous philosopher Kuno Fischer ( 1824–1907 ) (history of 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 ( 1833–1872 ) ( analytical geometry ), Ernst Christian Julius Schering ( 1824–1897 ) ( function theory ), Wilhelm Eduard Weber ( 1804–1891 ) ( physical studies, applied physics ), Eduard Riecke ( 1845–1915 ) ( theory of electricity ), and Rudolf Hermann Lotze ( 1817–1881 ) ( 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 Schering, with a dissertation under the title of "Über 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 Feb 1856 - 25 June 1904 ) on March 14, 1887.

Work as a logician

Main article: Begriffsschrift

Though his education and early work were mathematical, especially geometrical, Frege's thought soon turned to logic. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens ( Halle a/S: Verlag von Louis Nebert, 1879 ) ( 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 wanted to show that mathematics grows out of logic, but 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.

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 pair of sentences like "every boy loves some girl" and "some girl is loved by every boy" was able to 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".

It is frequently noted that Aristotle's logic is unable to represent even the most elementary inferences in Euclid's geometry, but Frege's "conceptual notation" can represent inferences involving indefinitely complex mathematical statements. The analysis of logical concepts and the machinery of formalization that is essential to Principia Mathematica ( 3 vols., 1910–1913 ) ( by Bertrand Russell and Alfred North Whitehead, 1861–1947 ), to Russell's theory of descriptions, to Kurt Gödel's ( 1906–1978 ) incompleteness theorems, and to Alfred Tarski's ( 1901–1983 ) 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 mathematical induction, were derived within what Frege understood to be pure logic.

This idea was formulated in non-symbolic terms in his Die Grundlagen der Arithmetik ( 1884 ) ( The Foundations of Arithmetic ). Later, in his Grundgesetze der Arithmetik ) ( vol. 1, 1893; vol. 2, 1903 ) ( Basic Laws of Arithmetic ) ( vol. 2 of which 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, if: ∃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 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".[1]

* Frege's logic, now known as second-order logic, can be weakened to so-called predicative second-order logic. However, this logic, although provably consistent by finitistic or constructive methods, can interpret only very weak fragments of arithmetic.

Frege's work in logic was little recognized in his day, in considerable part because his peculiar diagrammatic notation had no antecedents. It has since had no imitators. 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 ( 1815–1864 ) 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 ).

Philosopher

Frege is one of the founders of analytic philosophy, mainly because of his contributions to the philosophy of language, including the

* 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 ( born 1916 ) and Max Black ( 1909–1988 ), with the bibliographic assistance of Wittgenstein (see Geach, ed. 1975, Introduction). Hence, 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 Polish work on logic and semantics.

Sense and reference

Main article: Sense and Reference

The distinction between Sinn ( "sense" ) and Bedeutung ( usually translated "reference", but also as "meaning" or "denotation" ) was an innovation of Frege in his 1892 paper "Über Sinn und Bedeutung" ( "On Sense and Reference" ). According to Frege, sense and reference are two different aspects of the significance of an expression. Frege applied Bedeutung in the first instance to proper names, where it means the bearer of the name, the object in question, but then also to other expressions, including complete sentences, which bedeuten the two "truth values", the true and the false; 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.

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".

Imagine the road signs outside a city. They all point to ( bedeuten ) the same object ( the city ), although the "mode of presentation" or sense ( Sinn ) of each sign ( its direction or distance ) is different. Similarly "the Prince of Wales" and "Charles Philip Arthur George Mountbatten-Windsor" both denote ( bedeuten ) the same object, though each uses a different "mode of presentation" ( sense or Sinn ).

Political views and National Socialism

Frege was a sympathizer of the National Socialist party in its early stages[2], a known anti-Semite, and later in life named Adolf Hitler as one of his heroes[3]. For those who admire Frege's contributions to logic and philosophy, this may be a shock to learn, as this is often overlooked. As Reuben Hersh states in What Is Mathematics, Really? ( Oxford University Press, 1997 ), p. 241:

"Frege actually died a Nazi. Sluga reports: 'Frege confided in his diary in 1924 that he had once thought of himself as a liberal and was an admirer of Bismarck, but his heroes now were General Ludendorff and Adolf Hitler. This was after the two had tried to topple the elected democratic government in a coup in November 1923. In his diary Frege also used all his analytic skills to devise plans for expelling the Jews from Germany and for suppressing the Social Democrats.' Michael Dummett tells of his shock to discover, while reading Frege's diary, that his hero was an outspoken anti-Semite (1973)."[4]

Even prior to his sympathies with Hitler and the Nazis, Frege held very conservative political views. He disliked the small steps towards democracy made in the German Empire ( created 1871 ), not the least because it increased the power of the Socialists. His anti-Semitism fueled his desire to see all Jews expelled from Germany, or at least deprived certain political rights ( notwithstanding the fact that among his students was Gershom Scholem, 1897–1982, who much valued his teacher ). In addition to his anti-Semitic sentiments, his diaries also show a deep hatred of Catholics and of the French.[5][6]

Frege was described by his students as a highly introverted person, seldom entering the dialogue, mostly facing the blackboard while lecturing though being witty and sometimes bitterly sarcastic[7].

Important dates

* 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 — Professor Extraordinarius at Jena.

* 1896 — Ordentlicher Honorarprofessor at Jena.

* 1917 or 1918 — 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.

* English: Concept Notation, the Formal Language of the Pure Thought like that of Arithmetics.

Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl ( 1884 ). Breslau.

* English: The Foundations of Arithmetic: the logical-mathematical Investigation of the Concept of Number.

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

* English: Basic Laws of Arithmetic Vol. 1 ( 1893 ); Vol. 2 ( 1903 ). ISBN 978-0520047617

Philosophical studies

Function and Concept ( 1891 )

* Original: Funktion und Begriff : Vortrag, gehalten in der Sitzung; vom 9. Januar 1891 der Jenaischen Gesellschaft für Medizin und Naturwissenschaft, Jena, 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: "Über 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. Februar 1904, S. Meyer ( ed. ), Leipzig, 1904, pp. 656–666;

* In English: "What is a Function?"

Logical Investigations ( 1918–1923 ). Frege intended that the following three papers be published together in a book titled Logische Untersuchungen ( Logical Investigations ). Though the German book never appeared, English translations did appear together in Logical Investigations, ed. Peter Geach, Blackwell's, 1975.

* 1918–19. "Der Gedanke: Eine logische Untersuchung" ( "Thought: A Logical Investigation" ), in Beiträge zur Philosophie des Deutschen Idealismus I: 58–77.

* 1918–19. "Die Verneinung" ( "Negation" ) in Beiträge zur Philosophie des deutschen Idealismus I: 143–157.

* 1923. "Gedankengefüge" ( "Compound Thought" ), in Beiträge zur Philosophie des Deutschen Idealismus III: 36–51.

Articles on geometry

* 1903: "Über die Grundlagen der Geometrie". II. Jaresbericht der deutschen Mathematiker-Vereinigung XII ( 1903 ), 368–375;

o In English: "On the Foundations of Geometry".

* 1967: Kleine Schriften. ( I. Angelelli, ed. ) Wissenschaftliche Buchgesellschaft. Darmstadt, 1967 és G. Olms, Hildescheim, 1967. "Small Writings," a collection of most of his writings ( e.g., the previous ), posthumously published.

References

1. ^ Frege's Logic, Theorem, and Foundations for Arithmetic, Stanford Encyclopedia of Philosophy at plato.stanford.edu

2. ^ Hersh, Reuben, What Is Mathematics, Really? ( Oxford University Press, 1997 ), p. 241: IVS_2gMM7GlAeFyKS8BQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CBAQ6AEwAg#v=onepage&q=&f=false

3. ^ Hicks, Stephen, "Gottlob Frege and Naziism": http://www.stephenhicks.org/tag/gottlob-frege-and-nazism/

4. ^ Hersh, Reuben, What Is Mathematics, Really? ( Oxford University Press, 1997 ), p. 241: 5. ^ http://www-history.mcs.st-andrews.ac.uk/~history/Biographies/Frege.html

6. ^ http://www.iep.utm.edu/f/frege.htm

7. ^ Frege's Lectures on Logic, Erich H. Reck and Steve Awody, pp.18-26

Primary

* Online bibliography of Frege's works and their English translations.

* 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. "Über 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. Partial translation: Furth, M, 1964. The Basic Laws of Arithmetic. Univ. of California 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 ).

* 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:

* 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.

* Diamond, Cora, 1991. The Realistic Spirit. MIT Press. — Primarily about Wittgenstein, but contains several articles on Frege.

* 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.

* Smith, Leslie, 1999. "What Piaget Learned from Frege." Developmental Review 19(1): 133-153. — On why Frege first appears in Piaget's writings in 1949, twenty-five years after he began publishing on logic and epistemology.

* 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.

* 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, Douglas A., 1982. Frege, Dedekind, and Peano on the foundations of arithmetic. Assen, Netherlands: Van Gorcum.

* Charles Parsons, 1965, "Frege's Concept of Number." Reprinted with Postscript in Demopoulos ( 1965 ): 182-210. The starting point of the ongoing sympathetic reexamination of Frege's logicism.

* 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.

External links

* A comprehensive guide to Fregean material available on the web by Brian Carver.

* Stanford Encyclopedia of Philosophy:

o "Gottlob Frege" — by Edward Zalta.

o "Frege's Logic, Theorem, and Foundations for Arithmetic" — by Edward Zalta

* Internet Encyclopedia of Philosophy:

o Gottlob Frege — by Kevin C. Klement.

o 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, http://www-history.mcs.st-andrews.ac.uk/Biographies/Frege.html .

* Begriff, a LaTeX package for typesetting Frege's logic notation.

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