Gottlob Frege

Introduction

Gottlob Frege is the German mathematician and philosopher whose work in the late nineteenth and early twentieth centuries founded modern formal logic, founded the analytic philosophy of language, and decisively shaped the analytic philosophy of mathematics. His 1879 Begriffsschrift (Concept-Script) introduced the formal logical apparatus — quantification, the truth-functional account of connectives, the formal treatment of relations — that constitutes modern predicate logic and that almost no subsequent work in formal logic has altered. His 1892 paper Über Sinn und Bedeutung (On Sense and Reference) introduced the distinction between linguistic sense (Sinn) and reference (Bedeutung) that structures the analytic philosophy of language to this day. His larger project of deriving arithmetic from pure logic (logicism) was the founding research program of the analytic philosophy of mathematics.

Frege's contemporary recognition was modest — the Begriffsschrift was poorly received at publication; his logicist project collapsed when Bertrand Russell discovered the paradox that bears his name in 1902; he died in 1925 unrecognized. The posthumous reception has been the inverse: Frege is now generally regarded as one of the founders of analytic philosophy and one of the most important figures in the history of logic.

Life

Frege was born in 1848 in Wismar, a Baltic port in northern Germany, to a family of academics; his father was the headmaster of a girls' school he eventually inherited. He took his early education at the Wismar Gymnasium, then studied mathematics, physics, chemistry, and philosophy at Jena and Göttingen. He took his PhD in mathematics at Göttingen in 1873 with a dissertation on geometrical representation in the plane, and his Habilitation at Jena in 1874 with a thesis on calculation methods.

Frege spent his entire academic career at the University of Jena, where he taught from 1874 until his retirement in 1918. He was promoted slowly: Privatdozent in 1874, extraordinary professor (an unsalaried position) in 1879, honorary ordinary professor (still unsalaried by the university but supported by a grant) in 1896, and full professor only in 1903. The Jena career was institutionally marginal and personally difficult; Frege was not a successful teacher (most of his lectures attracted a handful of students or none) and was largely isolated from the mainstream of European mathematics and philosophy.

The major works appeared at intervals. The Begriffsschrift in 1879. Die Grundlagen der Arithmetik (The Foundations of Arithmetic) in 1884, the most accessible single statement of the logicist program. The two volumes of Die Grundgesetze der Arithmetik (The Basic Laws of Arithmetic) in 1893 and 1903, the systematic formal derivation of arithmetic from logic. The famous papers Function and Concept (1891), On Sense and Reference (1892), and On Concept and Object (1892), constituting the major philosophical writings on language and logic.

The logicist project collapsed in June 1902, when Frege received a letter from Bertrand Russell pointing out a contradiction in the axioms of Grundgesetze volume 1: the set of all sets that are not members of themselves both is and is not a member of itself (Russell's paradox). Frege's response, included as an appendix to Grundgesetze volume 2 (already at press), attempted a repair but could not preserve the systematic project. The collapse effectively ended Frege's productive work in the foundations of mathematics; the late writings (the Logical Investigations of 1918–1923) are but more philosophical than systematic.

Frege's personal life was difficult. He lost both his wife and his only son in adulthood; he raised an adopted son; his political views in the last decade of his life moved toward antisemitism and right-wing nationalism that have complicated his reception (his late diary entries are explicit; his earlier work shows no such tendencies). He died in 1925 in Bad Kleinen, near his birthplace.

The problem he worked on

Frege's central project was the systematic articulation of a rigorous foundation for arithmetic. Mathematics in the nineteenth century had undergone foundational work — Bolzano, Cauchy, Weierstrass had rigorized the calculus; Dedekind and Cantor had developed the foundations of analysis and set theory; the question of what number was and what justified arithmetical reasoning remained open. Frege's answer is logicism: arithmetic is (a part of) logic; numbers can be defined in purely logical terms; arithmetical theorems can be derived from purely logical axioms.

The project required the development of a formal logical apparatus adequate to the derivation. Existing logic — the Aristotelian syllogistic, Boolean propositional algebra — was inadequate to the relational and quantificational reasoning arithmetic required. The Begriffsschrift was Frege's invention of the formal apparatus needed for the project; the Grundlagen was the informal statement of the logicist program; the Grundgesetze was the formal execution.

When the project collapsed in 1902, the logical apparatus survived and was eventually shown to be correct in its representational structure even though Frege's specific axioms for set theory had to be replaced. The collapse of logicism in the strong form Frege intended is one of the major events in the foundations of mathematics; the survival of the logical apparatus is one of the major continuities in modern philosophy.

Contributions

Modern formal logic

The Begriffsschrift (1879) introduced quantification, the truth-functional account of propositional connectives, the formal treatment of relations, and the rigorous notation for representing complex propositions. The Fregean apparatus is, with notational modifications, the modern predicate calculus. Almost no subsequent work in formal logic has altered the basic representational framework Frege introduced; the developments of the twentieth century (Russell and Whitehead's Principia, the metalogical work of Gödel, Tarski, and Church, the modal extensions by Kripke and others) have built on rather than replaced the Fregean foundations.

The historical question of priority is complicated by the parallel work of Peirce in the United States. Peirce's 1880 paper On the Algebra of Logic and the 1885 paper introducing quantifier notation were partly independent and partly anticipated by Frege; the standard story of Frege as sole inventor of modern logic is an oversimplification, though Frege's work was systematically more developed and more rigorously axiomatized.

The sense-reference distinction

The 1892 paper On Sense and Reference introduced the distinction between linguistic sense (Sinn, the way an object is given to us) and reference (Bedeutung, the object itself). The motivation: the morning star is the evening star is an informative astronomical discovery, but the morning star is the morning star is a trivial tautology; if both sentences have the same constituents and the constituents have only reference, the difference in informativeness cannot be explained. Frege's solution: the constituents morning star and evening star have the same reference (the planet Venus) but different senses (the way Venus is given as the brightest object in the morning sky vs. as the brightest object in the evening sky); the informativeness of the identity is explained by the difference in senses.

The distinction has been one of the most influential single moves in the philosophy of language. It is presupposed by Russell's theory of descriptions, by Carnap's intensional logic, by Kripke's causal theory of reference (as the foil against which Kripke argues), by the contemporary literature on indexicals, demonstratives, propositional attitudes, and many other topics.

Logicism

The Grundlagen (1884) and the Grundgesetze (1893, 1903) develop the systematic derivation of arithmetic from purely logical principles. Numbers are defined in terms of extensions of concepts (the number 2 is the extension of the concept concept under which exactly two objects fall); arithmetical operations are defined as operations on these extensions; the axioms of arithmetic are derived from purely logical principles together with these definitions.

The program was demolished by Russell's paradox but the technical machinery survived. Frege's neo-logicism has been actively revived in the contemporary literature, especially through the work of Crispin Wright (Frege's Conception of Numbers as Objects, 1983), Bob Hale, and others, who argue that a modified version of the Fregean program can be carried through using Hume's Principle (the principle that the number of Fs equals the number of Gs if and only if there is a one-to-one correspondence between the Fs and the Gs) without the inconsistent axioms that doomed the original.

Concept and object

Frege's distinction between concepts (functions whose values are truth-values) and objects (the values such functions can take and the saturated entities such functions can apply to) is one of the more technical Fregean doctrines and one of the more philosophically contested. The distinction is essential to the formal apparatus of the Begriffsschrift and to Frege's account of arithmetic; it produces the famous puzzle that the concept horse is not a concept (because the concept horse is grammatically the name of an object, but if it referred to a concept, it would itself be a concept rather than an object).

The contemporary literature continues to engage the distinction, both as a technical matter (in the philosophy of mathematics and formal semantics) and as a philosophical matter (in the broader theory of predication and property).

Key works

• Begriffsschrift (1879). The invention of modern formal logic.
• Die Grundlagen der Arithmetik (1884). The informal statement of the logicist program.
• Function and Concept (1891). The technical paper on the function-object distinction.
• On Sense and Reference (1892). The founding paper of analytic philosophy of language.
• On Concept and Object (1892). The companion paper on the concept-object distinction.
• Die Grundgesetze der Arithmetik (volume 1, 1893; volume 2, 1903). The formal derivation of arithmetic, collapsed by Russell's paradox.
• Logical Investigations (three essays, 1918–1923, late period).

The standard scholarly edition is Frege's Nachgelassene Schriften edited by Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach (Hamburg, 1969); the standard English collection is The Frege Reader edited by Michael Beaney (Blackwell, 1997). The recent translation of Grundgesetze by Philip Ebert and Marcus Rossberg (Oxford, 2013) is the dominant English text.

Influences and influenced

Influenced by: Kant (whom Frege engaged on the nature of arithmetic in the Grundlagen); Leibniz (whose characteristica universalis anticipated Frege's logical project); the nineteenth-century German mathematical tradition (Weierstrass, Cantor, Dedekind); Hermann Lotze (who taught Frege at Göttingen).

Influenced: Bertrand Russell (who recognized the importance of Frege's work in the 1900s and whose Principles of Mathematics, 1903, was influenced by it); Ludwig Wittgenstein (whose Tractatus engages Frege extensively); the Vienna Circle (Carnap was a serious student of Frege); the entire analytic tradition's distinctive concerns with logic, language, and the foundations of mathematics; the contemporary neo-Fregean tradition through Crispin Wright, Bob Hale, and others.

Reception

The contemporary reception was modest. The Begriffsschrift was reviewed harshly or ignored; the Grundlagen attracted little attention; the Grundgesetze volume 1 sold poorly and the appearance of volume 2 with the Russell paradox appendix produced no response. Frege's recognition began with Russell's serious engagement with his work after 1900 and continued through Wittgenstein's Tractatus (1921), but Frege himself died largely unrecognized.

The twentieth-century recovery began in the 1950s with Michael Dummett's work, especially Frege: Philosophy of Language (1973) and Frege: Philosophy of Mathematics (1991), which established the contemporary scholarly framework for engaging Frege. The neo-Fregean program through Wright, Hale, and others has produced a contemporary research program. The history of analytic philosophy as a discipline has reframed itself around Frege's foundational role.

Continuing engagement

Major recent scholarly work includes Michael Dummett's two volumes (1973, 1991), Joan Weiner's Frege in Perspective (1990) and Frege (1999), Tyler Burge's collected essays Truth, Thought, Reason: Essays on Frege (2005), Richard Heck's Frege's Theorem (2011), and the work of Bob Hale, Crispin Wright, William Demopoulos, and Marcus Rossberg. Active scholarly debates concern the precise reading of the sense-reference distinction, the viability of the neo-Fregean program, the relation between Frege's formal apparatus and his philosophical motivations, and the reception history of the Begriffsschrift.

Further reading

• Analytic Philosophy — the tradition Frege founded
• Russell — the successor who discovered the paradox and developed the Fregean program
• Wittgenstein — the successor whose Tractatus engages Frege extensively
• Kant — the philosophical predecessor whose account of arithmetic Frege engaged
• Peirce — the parallel American figure whose logical work was partly independent

The German mathematician and philosopher who invented modern formal logic, founded the analytic philosophy of language with the sense-reference distinction, and founded the analytic philosophy of mathematics with the logicist program.