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Gottlob Frege

 
Britannica Concise Encyclopedia: Friedrich Ludwig Gottlob Frege

(born Nov. 8, 1848, Wismar, Mecklenburg-Schwerin — died July 26, 1925, Bad Kleinen, Ger.) German mathematician and logician, inventor of modern mathematical logic and one of the founders of the analytic tradition in philosophy. He taught at the University of Jena from 1871 to 1917. His Begriffsschrift (1879, "Conceptscript"), was the first presentation of a system of mathematical logic in the modern sense. Using an original notation of quantifiers and variables, he was able to give formal expression to sentences containing multiple quantification, such as "Everybody loves someone"; this is impossible in the syllogistic derived from Aristotle, which had been considered complete until the time of Immanuel Kant (see predicate calculus). In the Die Grundlagen der Arithmetik he attempted to establish the doctrine later known as logicism. He also made significant contributions to the philosophy of language, including a highly influential theory of the distinction between sense and reference. Though Frege's work was admired by Bertrand Russell and the early Ludwig Wittgenstein, it was unknown to or ignored by most other philosophers and mathematicians during Frege's lifetime; its significance was not generally appreciated until the mid-20th century.

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Scientist: Friedrich Ludwig Gottlob Frege
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German philosopher and mathematician (1848–1925)

Born at Wismar (now in Germany), Frege studied at the universities of Jena and Göttingen, where he obtained his PhD in 1873. He then returned to Jena as a lecturer, where he remained for the rest of his working life, rising to the position of professor in 1896. In a series of seminal works Frege laid the foundations of modern mathematical logic, transforming logic with an understanding of and notation for the problem of multiple generality – propositions containing predicates, quantifiers, and variables – and showing how the basic concepts and operations of mathematics could be formalized. He also revolutionized modern philosophy through his influence on the philosophy of language. However Frege's work was almost completely ignored, misunderstood, or treated with hostility by his contemporaries – notable exceptions were Bertrand Russell and Giuseppe Peano.

In his first major work, Begriffsschrift (Concept Writing, 1879), he provided a new formalism containing an adequate symbolism and an axiomatic base for the rigorous derivation of both propositional and predicate logic. While a few workers, such as the American C.S. Pierce, had been moving in this direction, their work was completely overshadowed by the comprehensive nature of Frege's work.

In Die Grundlagen der Arithmetik (1884; The Foundations of Arithmetic) Frege gave a formal definition of cardinal number and showed how basic properties of numbers could be logically derived from it. In Grundgesetze der Arithmetik (1893 and 1903; Basic Laws of Arithmetic) he went further in attempting to derive arithmetic from formal logic. The Grundgesetze is still regarded as a massive achievement, but his main aim was doomed to failure. On the eve of the publication of the second volume Russell wrote to Frege pointing out a contradiction – Russell's paradox – that could be derived from his system. This, as Frege acknowledged, vitiated his whole project.

Biography: Gottlob Frege
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The German mathematician and philosopher Gottlob Frege (1848-1925) is considered the founder of modern mathematical logic. His work was almost wholly ignored during his lifetime but now exerts a great influence on the philosophy of logic and language.

Gottlob Frege was born on Nov. 8, 1848, at Wismar. He began his university studies at Jena in 1869 but after 2 years moved to Göttingen. He studied mathematics, the natural sciences, and philosophy and took his degree in 1873. Thereafter he taught at Jena in the department of mathematics. He was made a professor in 1896 and retired in 1918. Frege was married to Margarete Lieseberg, and the couple had one adopted son. Frege died on July 26, 1925, in Bad Kleinen.

Frege invented the concept of a formal system of mathematical logic, and in his first major work, Begriffsschrift (1879), he presented the first example of such a system in his formulation of a propositional and predicate calculus. He introduced the mathematical notion of function and variable into logic and invented the idea of quantifiers. He was also the first writer on axiomatic theory to make clear the distinction between an axiom and a rule of inference.

Further progress in this work convinced Frege that the basic ideas of arithmetic (but not of geometry) could be articulated solely in logical expressions. He expressed his new program first in a nonsymbolic work, The Foundations of Arithmetic (1884), which also featured a brilliant and devastating polemic against all previous attempts at the subject. The crown of his work was to be his Basic Laws of Arithmetic. The first volume of this work appeared in 1893; but in 1903, as Frege was about to issue the second volume, Bertrand Russell pointed out a contradiction in Frege's use of the concept of a "class," which undermined the proofs in the work. Frege hastily added an appendix that sought to remedy the defect (this effort was later proved defective), but thereafter he seemed to lose interest in the great project. Two decades later he regarded the whole enterprise as an error and fell back upon the Kantian interpretation of mathematical judgments as synthetic a priori.

Frege also made important contributions to the philosophy of logic. Concerning the old question: what is it for a proposition to have meaning? - he introduced a variety of distinctions that are being exploited by contemporary philosophers. Frege rejected epistemology as the starting point of philosophy and revived the classical view, dominant before René Descartes, that philosophical logic holds this place.

Further Reading

There are no biographies of Frege. His work, however, has been extensively studied, especially since translations of it have become available, beginning in the 1950s. A convenient collection of most of the important critical essays is in E. D. Klemke, ed., Essays on Frege (1968), which also has a complete bibliography. Two difficult but rewarding full-length studies are Jeremy D. B. Walker, A Study of Frege (1965), and Robert Sternfeld, Frege's Logical Theory (1966).

Philosophy Dictionary: Gottlob Frege
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Frege, Gottlob (1848-1925) German mathematician and philosopher of mathematics. Frege was born in the small town of Wismar in Pomerania, and was sent to the university of Jena when he was twenty-one. He obtained his doctorate at Göttingen, and worked almost the whole of his life in the mathematics department at the university of Jena. His first important work, the Begriffsschrift (‘concept writing’ 1879), is also the first example of a formal system in the sense of modern logic. In it Frege undertakes to devise a formal system within which mathematical proofs may be given. It was his discovery of the correct representation of generality, the notion of quantifier and variable, that opened the possibility of successfully achieving this aim. With that notation Frege could represent sentences involving multiple generality (such as the form ‘for every small number e there is a number n such that…’) on which the validity of much mathematical reasoning depends. The Begriffsschrift also contains the elements of the propositional calculus, including an informal presentation of the notion of a truth-function. It is universally acknowledged to mark the beginning of modern logic. In 1884 Frege published the Grundlagen der Arithmetik (trs. as The Foundations of Arithmetic by J. L. Austin, 1959). In this work, Frege brilliantly attacks rival accounts of the status of arithmetic, and then propounds his own approach to the subject, analysing the basic concepts of mathematics in such a form that a reduction of arithmetic to operations that are fundamentally logical in nature becomes a real possibility. The first volume of the Grundgesetze der Arithmetik (1893, trs. as The Basic Laws of Arithmetic, 1964) formalizes the mathematical approach of the Grundlagen, a task that necessitated giving the first formal theory of classes; it was this theory that was later shown inconsistent by Russell's paradox. Volume ii of the Grundgesetze, concerned mainly with the theory of real numbers, was published in 1903. Frege's own reaction to Russell's paradox, after understandable initial consternation, was to modify one of his own axioms; the result, however, was not eventually tenable, and it was only with Zermelo's work that the modern conception of set theory was put on a satisfactory footing.

Frege's distinction as a logician is matched by his deep concern with the basic semantic concepts involved in the logical foundations of his work. In a succession of papers he forges the basic concepts and distinctions that have dominated subsequent philosophical investigation of logic and language. The topics of these writings include sense (Sinn) and reference, concepts, functions and objects, identity, negation, assertion, truth/falsity, and the nature of thought. Although his relations to the philosophical surroundings of his time are debatable, these concerns and his approach to them stamp Frege as the founding figure of analytical philosophy. However, his concern to protect a timeless objectivity for thought and its contents has led to accusations of Platonism, and his own views of the objects of mathematics troubled him until the end of his life. Translations include Translations from the Philosophical Writings of Gottlob Frege, edited by P. Geach and M. Black (1960), The Basic Laws of Arithmetic, translated and edited by M. Furth (1964), Conceptual Notation and Related Articles, edited by T. W. Bynum (1972), and On the Foundations of Geometry and Formal Theories of Arithmetic, edited by E.-H. W. Kluge (1971).

 
Columbia Encyclopedia: Gottlob Frege
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Frege, Gottlob (gôt'lōp frā'), 1848-1925, German philosopher and mathematician. He was professor of mathematics (1879-1918) at the Univ. of Jena. Frege was one of the founders of modern symbolic logic, and his work profoundly influenced Bertrand Russell. He claimed that all mathematics could be derived from purely logical principles and definitions. He considered verbal concepts to be expressible as symbolic functions with one or more variables. His books include Begriffsschrift (1879); Die Grundlagen der Arithmetik (1884; tr. The Foundations of Arithmetic, 1950); Grundgesetze der Arithmetik (2 vol., 1893-1903).

Bibliography

See P. T. Geach and M. Black, ed., Philosophical Writings of Gottlob Frege (1952); M. Resnik, Frege and the Philosophy of Mathematics (1980); M. Dummett, The Interpretation of Frege's Philosophy (repr. 1981).

Wikipedia: Gottlob Frege
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Gottlob Frege
Western Philosophy
19th century philosophy
Full name Friedrich Ludwig Gottlob Frege
Born 8 November 1848
Wismar, Germany
Died 26 July 1925 (aged 76)
Bad Kleinen, Germany
School/tradition Analytic philosophy
Main interests Philosophy of mathematics, mathematical logic, Philosophy of language
Notable ideas Principle of compositionality, Predicate calculus, Logicism, Sense and reference

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, and later Bertrand Russell who helped introduce his work to the later generations of logicians and philosophers.

Contents

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, Karl Alexander Frege, was the founder and headmaster of a girls' high school until his death in 1866. Afterwards, the school was led by Frege's mother, Auguste Wilhelmine Sophie Frege (née Bialloblotzky, apparently 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, the first section of which dealt with the structure and logic of language.[citation needed]

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.

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 Abbe (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 Zeiss, 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 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 Christian Julius 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 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.

Work as a logician

Gottlob Frege

Though his education and early work were mathematical, especially geometrical, Frege's thought soon turned to logic. His 1879 Begriffsschrift (Concept Script) 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 grew out of logic, but in so doing 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, he invented axiomatic predicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and 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 Bertrand Russell's theory of descriptions and Principia Mathematica (with Alfred North Whitehead), and to Gödel's incompleteness theorems, and to Alfred Tarski's 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 Foundations of Arithmetic of 1884. Later, in the Basic Laws of Arithmetic (Grundgesetze der Arithmetik (1893, 1903)), published at its author's 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[FxGx]. The set of Fs is the same as the set of Gs just in case every F is a G and every G is an F. (The case is special because what is here being called the extension of a predicate, or a set, is only one type of "value-range" of a function.)

In a famous episode, Bertrand Russell wrote to Frege, just as Vol. 2 of the Grundgesetze was about to go to press in 1903, showing that Russell's paradox could be derived from Frege's Basic Law V. It is easy to define the relation of membership of a set or extension in Frege's system; Russell then drew attention to "the set of things x that are such that x is not a member of x". The system of the Grundgesetze entails that the set thus characterised both is and is not a member of itself, and is thus inconsistent. Frege wrote a hasty, last-minute Appendix to Vol. 2, deriving the contradiction and proposing to eliminate it by modifying Basic Law V. (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 George Boolos. 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 & ∀xy(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(FxGx). 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 Principia Mathematica appeared in 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.

Philosopher

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

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 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.

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 the famous lectures on "Naming and Necessity" of Saul Kripke.

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

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. He is known to have held antisemitic views, such as the 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, who much valued his teacher). His diaries also show a deep hatred of Catholics and of the French.[2][3]

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[4].

Important dates

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.

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

  • English: Basic Laws of Arithmetic Vol. 1 (1893); Vol. 2 (1903).

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)

"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;
    • 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

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. 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).

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.

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