Gottlob Frege

Though his education and early mathematical work focused primarily on geometry, Frege's work soon turned to logic. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens [Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic], Halle a/S: Verlag von Louis Nebert, 1879 marked a turning point in the history of logic. The Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables. Frege's goal was to show that mathematics grows out of logic, and in so doing, he devised techniques that took him far beyond the Aristotelian syllogistic and Stoic propositional logic that had come down to him in the logical tradition.

Title page to Begriffsschrift (1879)

In effect, Frege invented axiomatic predicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality. Previous logic had dealt with the logical constants and, or, if... then..., not, and some and all, but iterations of these operations, especially "some" and "all", were little understood: even the distinction between a sentence like "every boy loves some girl" and "some girl is loved by every boy" could be represented only very artificially, whereas Frege's formalism had no difficulty expressing the different readings of "every boy loves some girl who loves some boy who loves some girl" and similar sentences, in complete parallel with his treatment of, say, "every boy is foolish".

A frequently noted example is that Aristotle's logic is unable to represent mathematical statements like Euclid's theorem, a fundamental statement of number theory that there are an infinite number of prime numbers. Frege's "conceptual notation", however, can represent such inferences.[19] The analysis of logical concepts and the machinery of formalization that is essential to Principia Mathematica (3 vols., 1910–13, by Bertrand Russell, 1872–1970, and Alfred North Whitehead, 1861–1947), to Russell's theory of descriptions, to Kurt Gödel's (1906–78) incompleteness theorems, and to Alfred Tarski's (1901–83) theory of truth, is ultimately due to Frege.

One of Frege's stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to "intuition". If there was an intuitive element, it was to be isolated and represented separately as an axiom: from there on, the proof was to be purely logical and without gaps. Having exhibited this possibility, Frege's larger purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism: unlike geometry, arithmetic was to be shown to have no basis in "intuition", and no need for non-logical axioms. Already in the 1879 Begriffsschrift important preliminary theorems, for example, a generalized form of law of trichotomy, were derived within what Frege understood to be pure logic.

This idea was formulated in non-symbolic terms in his The Foundations of Arithmetic (Die Grundlagen der Arithmetik, 1884). Later, in his Basic Laws of Arithmetic (Grundgesetze der Arithmetik, vol. 1, 1893; vol. 2, 1903; vol. 2 was published at his own expense), Frege attempted to derive, by use of his symbolism, all of the laws of arithmetic from axioms he asserted as logical. Most of these axioms were carried over from his Begriffsschrift, though not without some significant changes. The one truly new principle was one he called the 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, that is, the set of all Fs, and similarly for Gx. Then Basic Law V says that the predicates Fx and Gx have the same extension if and only if ∀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, unless: ∃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".[20]
• Frege's logic, now known as second-order logic, can be weakened to so-called predicative second-order logic. Predicative second-order logic plus Basic Law V is provably consistent by finitistic or constructive methods, but it can interpret only very weak fragments of arithmetic.[21]

Frege's work in logic had little international attention until 1903 when Russell wrote an appendix to The Principles of Mathematics stating his differences with Frege. The diagrammatic notation that Frege used had no antecedents (and has had no imitators since). 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 (1815–64) 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).

Frege is one of the founders of analytic philosophy, whose work on logic and language gave rise to the linguistic turn in philosophy. His contributions to the philosophy of language include:

• Function and argument analysis of the proposition;
• Distinction between concept and object (Begriff und Gegenstand);
• Principle of compositionality;
• Context principle; and
• 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 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 (1916–2013) and Max Black (1909–88), with the bibliographic assistance of Wittgenstein (see Geach, ed. 1975, Introduction). 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 work on logic and semantics by Polish logicians.

Frege's 1892 paper, "On Sense and Reference" ("Über Sinn und Bedeutung"), introduced his influential distinction between sense ("Sinn") and reference ("Bedeutung", which has also been translated as "meaning", or "denotation"). While conventional accounts of meaning took expressions to have just one feature (reference), Frege introduced the view that expressions have two different aspects of significance: their sense and their reference.

Reference (or "Bedeutung") applied to proper names, where a given expression (say the expression "Tom") simply refers to the entity bearing the name (the person named Tom). Frege also held that propositions had a referential relationship with their truth-value (in other words, a statement "refers" to the truth-value it takes). By contrast, the sense (or "Sinn") associated with a complete sentence is the thought it expresses. The sense of an expression is said to be the "mode of presentation" of the item referred to, and there can be multiple modes of representation for the same referent.

The distinction can be illustrated thus: In their ordinary uses, the name "Charles Philip Arthur George Mountbatten-Windsor", which for logical purposes is an unanalyzable whole, and the functional expression "the Prince of Wales", which contains the significant parts "the prince of 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".

Frege's published philosophical writings were of a very technical nature and divorced from practical issues, so much so that Frege scholar Dummett expresses his "shock to discover, while reading Frege's diary, that his hero was an anti-Semite."[22] After the German Revolution of 1918–19 his political opinions became more radical. In the last year of his life, at the age of 76, his diary contained political opinions opposing the parliamentary system, democrats, liberals, Catholics, the French and Jews, who he thought ought to be deprived of political rights and, preferably, expelled from Germany.[23] Frege confided "that he had once thought of himself as a liberal and was an admirer of Bismarck", but then sympathized with General Ludendorff. Some interpretations have been written about that time.[24] The diary contains a critique of universal suffrage and socialism. Frege had friendly relations with Jews in real life: among his students was Gershom Scholem,[25][26] who greatly valued his teaching, and it was he who encouraged Ludwig Wittgenstein to leave for England in order to study with Bertrand Russell.[27] The 1924 diary was published posthumously in 1994.[28] Frege apparently never spoke in public about his political viewpoints.

Frege was described by his students as a highly introverted person, seldom entering into dialogues with others and mostly facing the blackboard while lecturing. He was, however, known to occasionally show wit and even bitter sarcasm during his classes.[29]

• 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 — Ausserordentlicher Professor at Jena.
• 1896 — Ordentlicher Honorarprofessor at Jena.
• 1917 or 1918 — retires.
• Died 26 July 1925 in Bad Kleinen (now part of Mecklenburg-Vorpommern).

• Frege system
• List of pioneers in computer science
• Neo-Fregeanism

• Works by or about Gottlob Frege at Internet Archive
• Frege at Genealogy Project
• 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.
• O'Connor, John J.; Robertson, Edmund F., "Gottlob Frege", MacTutor History of Mathematics archive, University of St Andrews.
• Begriff, a LaTeX package for typesetting Frege's logic notation, earlier version.
• grundgesetze, a LaTeX package for typesetting Frege's logic notation, mature version
• Frege's Basic Laws of Arithmetic, website, incl. corrigenda and LaTeX typesetting tool — by P. A. Ebert and M. Rossberg.