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Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin – 26 July 1925, Bad Kleinen, Germany) was a German mathematician who became a logician and philosopher. He helped found both modern mathematical logic and analytic philosophy. His work had a far-reaching and foundational influence on 20th-century philosophy.

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

Frege studied at a gymnasium in Wismar, and graduated at the age of 15. His teacher Leo Sachse (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.

His other notable university teachers were Karl Snell (subjects: use of infinitesimal analysis in geometry, analytical geometry of planes, analytical mechanics, optics, physical foundations of mechanics); Hermann Schäffer (analytical geometry, applied physics, algebraic analysis, on the telegraph and other electronic machines); and the famous philosopher, Kuno Fischer (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 Alfred Clebsch (analytical geometry), Ernst Schering (function theory), Wilhelm Weber (physical studies, applied physics), Eduard Riecke (theory of electricity), and Rudolf Hermann Lotze (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 there was a direct influence arising from Frege's 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.

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 more ultimate purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism: unlike geometry it was to be shown to have no basis in "intuition," and no need on non-logical axioms. Already in the 1879 Begriffsschrift important preliminary theorems, for example a generalized form of mathematical induction, were derived within what he understood to be pure logic.

This idea was formulated in non-symbolic terms in his "Foundations of Arithmetic" of 1884. Later, in the "Basic Laws of Arithmetic" (Grundgesetze der Arithmetik (1893, 1903)), published at its author's expense, he attempted to derive all of the laws of arithmetic by use of his symbolism 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 both that it is and that it was not a member of itself, and was 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. (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 e.g. 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 George Boolos. A "concept" F is "small" if the objects falling under F cannot be put in 1-to-1 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, 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 Principia Mathematica appeared, 1910-13, the dominant approach to mathematical logic was still that of George Boole and his descendants, especially Ernst Schroeder. Frege's logical ideas nevertheless spread through the writings of his student Rudolf Carnap and other admirers, particularly Bertrand Russell and Ludwig Wittgenstein.

- 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 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 and Max Black, 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.

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).

- Born November 8, 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 — Ordenlicher Honorarprofessor at Jena.
- 1917 or 1918 — retires.
- Died July 26, 1925 in Bad Kleinen (now part of Mecklenburg-Vorpommern).

- 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).

- 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.

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 (in German): 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, Blackwells, 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.

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

- 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. Uni. 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).

- 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.

- A comprehensive guide to Fregean material available on the web by Brian Carver.
- Stanford Encyclopedia of Philosophy:
- " Gottlob Frege" — by Edward Zalta.
- " Frege's Logic, Theorem, and Foundations for Arithmetic" — by Edward Zalta
- Internet Encyclopedia of Philosophy:
- Gottlob Frege — by Kevin C. Klement.
- Frege and Language — by Dorothea Lotter.
- Metaphysics Research Lab: Gottlob Frege.
- Frege on Being, Existence and Truth.
- Begriff, a LaTeX package for typesetting Frege's logic notation.

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