Gottlob Frege

Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher who taught at the University of Jena for over forty years and was almost unknown during his lifetime. He is now regarded as the greatest logician since Aristotle and the founder of analytic philosophy (Analytic Philosophy). In the Begriffsschrift (“Concept-Script”, 1879) he invented modern quantificational logic, the logic of “every” and “some” with variables and functions, which made it possible for the first time to formalise the reasoning of mathematics. In The Foundations of Arithmetic (1884) he argued that arithmetic is reducible to logic — the programme called logicism — and in “On Sense and Reference” (1892) he drew the distinction between the sense (Sinn) and the reference (Bedeutung) of an expression that founded the modern philosophy of language. His system collapsed in 1902 when Bertrand Russell discovered a contradiction in it; but through Russell, Wittgenstein and Rudolf Carnap, who attended his lectures, Frege’s ideas became the foundation of twentieth-century logic, mathematics, linguistics and computer science.

Life

Frege was born on 8 November 1848 in Wismar, on the Baltic coast of Mecklenburg, where his father ran a girls’ school. He studied mathematics, physics, chemistry and philosophy at Jena and Göttingen, where he took his doctorate in 1873 with a dissertation on geometry. In 1874 he began teaching mathematics at the University of Jena, where he remained until his retirement in 1918, becoming an honorary (not a full) professor. His life was outwardly uneventful and saddened: his wife Margarete died in 1904, their own children died young, and he raised an adopted son, Alfred. His works were reviewed coolly or not at all; his notation, in which formulas branch across the page in two dimensions, was thought forbidding. In old age, embittered, he retired to Bad Kleinen near Wismar, where he died on 26 July 1925.

The Begriffsschrift

Since Aristotle, logic had been the theory of the syllogism — “All men are mortal; Socrates is a man; therefore Socrates is mortal” — which deals well with simple subject–predicate sentences but cannot represent the reasoning in mathematics, which constantly involves multiple generality: “for every number there is a greater number”, “every point is between some two points”. Frege’s Begriffsschrift (1879), subtitled “a formula language, modelled upon that of arithmetic, for pure thought”, solved the problem. Its key innovations were:

• Replacing the grammatical analysis of sentences into subject and predicate with the mathematical analysis into function and argument: “Socrates is mortal” consists of the function “x is mortal” applied to the argument Socrates.
• The quantifier with bound variables: “for every x…”, “there is some x such that…”, which can be nested to express any degree of generality.
• A complete treatment of the logical connectives (not, if… then) and a set of axioms and a rule of inference (modus ponens) from which theorems are derived with complete rigour, so that no step depends on intuition.

This was the birth of modern logic. With it, Frege could define ideas like the “ancestral” of a relation (the relation of being an ancestor, defined from the relation of being a parent), which he used to give a purely logical definition of mathematical induction.

The Foundations of Arithmetic

In Die Grundlagen der Arithmetik (1884), a short, brilliant, almost formula-free book, Frege asked: what is a number? He demolished existing answers. Numbers are not physical properties of things: the same pile can be “one pack” or “fifty-two cards”. They are not ideas in the mind (psychologism), for then there would be as many number twos as minds. They are not abstractions from experience, as John Stuart Mill had held. Against Immanuel Kant, who held that arithmetic, like geometry, is “synthetic a priori”, based on intuition, Frege argued that arithmetic is analytic, derivable from logic and definitions alone.

His positive answer: a statement of number is an assertion about a concept. “Jupiter has four moons” says that the concept moon of Jupiter has four instances. Numbers are objects — the number of Fs is the extension (class) of all concepts equinumerous with F — and two concepts are equinumerous if their instances can be paired one to one. (This criterion, now called Hume’s principle, is the core of the neo-logicist revival in recent decades.) The book also stated three principles: always separate the psychological from the logical; never ask for the meaning of a word in isolation, but only in the context of a sentence (the context principle); and never lose sight of the distinction between concept and object.

Sense and reference

In “Über Sinn und Bedeutung” (“On Sense and Reference”, 1892) Frege posed a puzzle. If “the morning star” and “the evening star” both name the planet Venus, why is “the morning star is the evening star” an astronomical discovery, while “the morning star is the morning star” is trivial? His answer was that expressions have both a reference (the object they pick out) and a sense (the “mode of presentation” of that object, the way it is given to thought). The two names have the same reference but different senses.

He extended the distinction to whole sentences: the sense of a sentence is a thought (Gedanke) — not a private idea, but an objective content that different people can grasp, share and assert — and its reference is its truth value, the True or the False. This explained why replacing a word with another of the same reference preserves truth in ordinary sentences but not inside reports of belief: “the ancients believed that the morning star was the evening star” may be false. The distinction remains at the centre of debates about meaning, reference, names and belief, revived and challenged in the twentieth century by Russell, Carnap, Kripke and many others.

The Basic Laws and the paradox

Frege’s great work, Grundgesetze der Arithmetik (“Basic Laws of Arithmetic”, vol. I 1893, vol. II 1903), set out to derive arithmetic from logic with complete formal rigour. One of its axioms, Basic Law V, said in effect that every concept determines a class (its extension). In June 1902 Russell wrote to him: consider the class of all classes that are not members of themselves; is it a member of itself? If it is, it is not; if it is not, it is. The axiom led to contradiction. Frege replied the same week, acknowledging that “arithmetic totters”, and added an appendix to volume II attempting a repair, which was later shown not to work. Russell admired his response: “As I think about acts of integrity and grace, I realise there is nothing in my knowledge to compare with Frege’s dedication to truth.”

Frege never fully recovered his project. In his last years he concluded that the source of arithmetic might be geometrical intuition after all. Yet the contradiction itself — Russell’s paradox — became one of the driving problems of twentieth-century mathematics, leading to axiomatic set theory, type theory and the work of Gödel.

Late essays

In the essays collected as Logical Investigations — “The Thought” (1918), “Negation” (1918), “Compound Thoughts” (1923) — Frege argued that thoughts belong to a “third realm”, neither physical things nor mental ideas, but objective and timeless, grasped but not created by thinking. Truth is not a property we produce but one we recognise. This anti-psychologism, which also influenced Edmund Husserl, made the study of meaning independent of psychology and set the course for the philosophy of language.

Legacy

Frege’s influence came late and mostly at second hand. Russell’s Principles of Mathematics (1903) included an appendix on his doctrines; Wittgenstein visited him in Jena and called his writing “the great works of Frege” in the preface of the Tractatus; Carnap attended his lectures. From the 1950s, the English philosopher Michael Dummett made Frege the hero of a history in which philosophy of language is “first philosophy”. Every programming language, every formal semantics of natural language, and every textbook of logic is built on the foundation he laid.

In conversation

• Russell’s letter — the paradox that shook the foundations of arithmetic.
• Morning star, evening star — the sense–reference distinction.
• Logicism — whether mathematics is nothing but logic.