Gottlob Frege

Gottlob Frege
Western Philosophy
19th-century philosophy,
Young frege.jpg
Friedrich Ludwig Gottlob Frege
Full name Friedrich Ludwig Gottlob Frege
Birth 8 November 1848
Death 26 July 1925
School/tradition Analytic philosophy
Main interests Philosophy of mathematics, mathematical logic, Philosophy of language
Notable ideas Predicate calculus, Logicism, Sense and reference
Influenced

Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin – 26 July 1925, Bad Kleinen, Germany) (IPA: [ˈgɔtlop ˈfʁeːgə]) 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 and linguistic semantics.

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 of a girls' high school, of which he was the headmaster 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.

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 there he attended approximately twenty courses of lectures, most of them on mathematics and physics. The teacher most important to him 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 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

Main article: Begriffsschrift
Gottlob Frege

Though his education and early work were mathematical, and 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 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, 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. (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:

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.

Philosopher

Main article: On Sense and Reference

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.

Sinn and Bedeutung

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 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).

Important dates

Important works

Logic; foundation of arithmetic

Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (1879). Halle a. S.

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.

Philosophical studies

Function and Concept (1891)

"On Sense and Reference" (1892)

"Concept and Object" (1892)

"What is a Function?" (1904)

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.

Articles on geometry

References

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

Primary

Secondary

Philosophy:

Logic and mathematics:

External links

Persondata
NAME Frege, Friedrich Ludwig Gottlob
ALTERNATIVE NAMES Frege, Gottlob
SHORT DESCRIPTION Important German logician and philosopher
DATE OF BIRTH November 8, 1848
PLACE OF BIRTH Wismar
DATE OF DEATH July 26, 1925
PLACE OF DEATH Bad Kleinen