Gottlob Frege

How to use this course

Gottlob Frege spent his whole career at a small university, and was read by a handful of people while he lived. He is now counted as the founder of modern logic and one of the founders of analytic philosophy. In 1879 he gave the first complete formal treatment of quantifiers and variables. He then tried to show that arithmetic is part of logic. The attempt failed when Bertrand Russell found a contradiction in his axioms in 1902, but the tools Frege built for it, and the philosophy of language he developed along the way, outlived the failure.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. We write Frege’s formulas in modern linear notation, with \(\forall\), \(\exists\), \(\to\) and \(\neg\), and show his own two-dimensional notation only in the hero figure and Lesson 04. Lessons 04 to 06 follow the Begriffsschrift (1879); Lesson 07 the Grundlagen (1884); Lesson 08 the papers of 1891–1892; Lesson 09 the Grundgesetze (1893, 1903); Lesson 10 the debate with David Hilbert. The companion courses on Bertrand Russell, Kurt Gödel and Alonzo Church continue the story.

Timeline

Wismar, Jena, Göttingen, 1848–1874

Friedrich Ludwig Gottlob Frege was born on 8 November 1848 in Wismar, a Baltic port in the Grand Duchy of Mecklenburg-Schwerin. His father, Carl Alexander Frege, had co-founded a private girls’ high school and was its headmaster. After his father’s death in 1866, his mother, Auguste, née Bialloblotzky, ran the school. Frege attended the Gymnasium in Wismar, where his teacher Leo Sachse is often credited with turning him toward mathematics, and left it in 1869.

He entered the University of Jena that spring. Over four semesters he took mathematics, physics, chemistry and philosophy. The physicist and optician Ernst Abbe, then a young lecturer and later the scientific force behind the Carl Zeiss works, became his most important teacher and, for decades, his protector. In 1871 Frege moved to Göttingen. There he heard Alfred Clebsch on analytic geometry, Ernst Schering on function theory, Wilhelm Weber on physics and Hermann Lotze on the philosophy of religion. How much Lotze shaped his later philosophy is still debated.

Two theses

Frege received his doctorate at Göttingen in 1873, with Schering as supervisor, for Über eine geometrische Darstellung der imaginären Gebilde in der Ebene, a study of how to represent imaginary points and curves of the complex projective plane by real geometric figures. A year later he habilitated at Jena with Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen, on methods of calculation based on an extension of the concept of magnitude. It treated iterated functions and functional equations. Read today, it shows two lasting interests: the generality of the notion of function, and the search for what is common to all quantities. In May 1874 he became a Privatdozent at Jena, paid only by student fees.

A concept-script, 1874–1884

For five years Frege taught a heavy load of mathematics courses and published little. Then, in 1879, he produced an 88-page book with an unusual title: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, a concept-script, a formula language of pure thought modelled on that of arithmetic. In it he set out to check whether arithmetic could be proved from logic alone. Ordinary language, he found, was too loose to show whether a chain of reasoning had gaps, so he built a language in which every inference could be checked.

The book contains the first formal system of quantificational logic: variables and quantifiers, a precise syntax, axioms and rules of inference. It also defines the ancestral of a relation, which yields a purely logical account of mathematical induction (Lesson 06). The same year, on Abbe’s recommendation, he was promoted to extraordinary professor.

The reception was cool. The two-dimensional notation looked strange, and reviewers who knew George Boole’s algebra of logic, among them Ernst Schröder and John Venn, judged the book a cumbersome rival to it. Frege replied in essays comparing the two approaches. Boole’s algebra, he argued, could express the relations among concepts but not the content of a mathematical statement. His own script did both. Charles Sanders Peirce and his student O. H. Mitchell reached a notation for quantifiers independently in 1883, and it was Peirce’s line, through Schröder and Giuseppe Peano, that most logicians followed for the next two decades.

The logicist programme, 1884–1903

Discouraged by the reaction to the formal book, Frege wrote the next one without formulas. Die Grundlagen der Arithmetik (1884) argued informally that numbers are objects, that statements of number are about concepts, and that the truths of arithmetic are analytic, contrary to Immanuel Kant. It dismantled the views of John Stuart Mill, who held numbers to be empirical, and of the formalists and psychologists of the day. Georg Cantor reviewed it unfavourably in 1885. Few others noticed it.

In 1887 Frege married Margarete Lieseberg. In the early 1890s he published three papers that are now classics of the philosophy of language: “Funktion und Begriff” (1891), “Über Sinn und Bedeutung” and “Über Begriff und Gegenstand” (both 1892). They were written to clarify the foundations of his logic before he presented it in full (Lesson 08).

The full presentation was Grundgesetze der Arithmetik. The first volume appeared in 1893 with the Jena publisher Hermann Pohle. In its preface Frege attacked the psychological logicians at length, which did not win him readers. The volume restated the logic with a new primitive, the course-of-values of a function, governed by an axiom that he called Basic Law V (Lesson 09). In 1896 he was named honorary ordinary professor, a title without the full rights of a chair.

Through the 1890s he corresponded with Peano on notation, reviewed Edmund Husserl’s Philosophie der Arithmetik severely (1894), and from 1895 exchanged letters with David Hilbert. When Hilbert’s Grundlagen der Geometrie appeared in 1899, Frege challenged its view of axioms (Lesson 10). By June 1902 the second volume of the Grundgesetze, which was to complete the derivation of the theory of real numbers, was in press.

Function, argument and quantifier

For the general reader. Before Frege, logic analysed sentences as subject and predicate: “Socrates is mortal”. That analysis cannot express “every number has a successor”, where one quantity depends on another. Frege replaced subject and predicate by function and argument, borrowed from mathematics. With variables and a sign for generality, he could express any statement of mathematics, including nested ones such as “for every \(\varepsilon\) there is a \(\delta\)”.

Function and argument

In the Begriffsschrift, an expression containing a replaceable part is split into a function (the fixed part) and an argument (the replaceable part). “Hydrogen is lighter than carbon dioxide” can be analysed with argument “hydrogen” and function “… is lighter than carbon dioxide”, or with argument “carbon dioxide”. Which split is chosen depends on the inference to be made. A function of one argument whose values are judgeable contents later became Frege’s concept; a function of two arguments, a relation.

The primitives

Frege took the conditional and negation as primitive connectives. In his notation the content stroke is a horizontal line; a vertical stroke at its left end, the judgement stroke, asserts the content. The conditional “if \(B\) then \(A\)” stacks \(A\) above \(B\) and joins them with a vertical line. Negation is a small vertical stroke hanging below the content line. Conjunction and disjunction are defined: \(A\wedge B\) as \(\neg(A\to\neg B)\) and \(A\vee B\) as \(\neg A\to B\).

Generality is marked by a concavity in the content line, with a German letter written in it. The scope of the quantifier is everything to the right of the concavity. This made Frege’s quantifier a variable-binding operator with explicit scope, the decisive innovation. The existential quantifier is defined as \(\neg\forall\neg\).

The gain is uniformity. Relations and multiple quantifiers are handled by the same rules as one-place predicates. For example the continuity of \(f\) at \(a\),

is a single formula whose logical form fixes which inferences are valid. Uniform continuity differs only in the order of the quantifiers \(\forall a\) and \(\exists\delta\), a distinction syllogistic logic cannot represent.

The first formal system

For the general reader. Frege wanted proofs in which nothing is taken for granted. He wrote down a small list of logical laws and two or three rules for passing from laws to new truths, then derived everything else step by step. This is what a formal system is today, and the Begriffsschrift is the first complete one for the logic of quantifiers.

Axioms (1) and (2) are the two standard axioms of today’s Hilbert-style propositional calculus, and with (28), (31) and (41) they give complete classical propositional logic; Jan Łukasiewicz later showed that (8) is derivable from (1) and (2). Axiom (58) and the generalization rule handle the quantifier. Frege allowed his letters to range over functions as well as objects, so his system is second-order. Restricted to first-order quantification, it is complete, as Gödel proved in 1929 for a closely related system.

A derivation

To derive \(a\to a\): substitute \(b\to a\) for \(b\) and \(a\) for \(c\) in (2); use (1) twice, once in the form \(a\to((b\to a)\to a)\) and once as \(a\to(b\to a)\):

The ancestral and induction

For the general reader. “Ancestor” is defined from “parent” by repetition: a parent, or a parent’s parent, and so on. The “and so on” seemed to need numbers, which would make any logical definition of number circular. Frege found a way to define the ancestor relation in one step, without counting. With it, mathematical induction stops being a special principle of arithmetic and becomes a theorem of logic.

The definition quantifies over all properties \(F\). This impredicative second-order quantification is what makes a one-step definition possible: \(y\) is a descendant of \(x\) if it has every hereditary property that the children of \(x\) have.

Proof. If \(F(x)\) and \(F\) is hereditary, then every \(z\) with \(f(x,z)\) has \(F\). So \(F\) satisfies the antecedent of the definition, and \(F(y)\) follows for any \(y\) with \(f^*(x,y)\). For transitivity, suppose \(f^*(x,y)\) and \(f^*(y,w)\), and let \(F\) be hereditary with \(F(z)\) for all \(f\)-successors \(z\) of \(x\). Then \(F(y)\), so all \(f\)-successors of \(y\) have \(F\) by heredity, and \(F(w)\) follows from \(f^*(y,w)\). The last claim is immediate from the definition.

The natural numbers

In the Grundlagen (§§79–83) Frege applied this to the successor relation \(S\): a natural number is anything that is \(0\) or follows \(0\) in the \(S\)-sequence, \(\mathbb N(n)\iff S^{*=}(0,n)\). Then ordinary induction is the general theorem applied to \(S\): if \(F(0)\) and \(F\) is hereditary in the \(S\)-sequence, every natural number has \(F\). Richard Dedekind reached an equivalent definition, through chains, in Was sind und was sollen die Zahlen? (1888), independently of Frege.

Numbers as objects: Hume’s principle

For the general reader. What is the number 4? Frege argued that a statement of number says something about a concept: “Jupiter has four moons”, as known in 1884, assigns the number four to the concept “moon of Jupiter”. Two concepts have the same number when their instances can be paired off one to one. From this single principle, and logic, all of arithmetic follows. That last fact was only fully appreciated a century later.

The arguments of the Grundlagen

Frege argued that numbers are not physical aggregates (the same pile is one pack or 52 cards), not mental ideas (arithmetic would then depend on psychology), and not properties of things. They are assigned to concepts, and number words function as singular terms, so numbers are objects. He insisted that one should ask for the meaning of a word only in the context of a sentence, the context principle. So to define a number term, one must explain the truth conditions of identity statements containing it.

The name comes from Frege’s citation of David Hume. Frege did not take the principle as a definition, because it does not settle whether, say, Julius Caesar is a number: the Caesar problem. He defined instead (§68) \(\#F\) as the extension of the concept “equinumerous with \(F\)”, and derived Hume’s principle from that. Extensions were later governed by Basic Law V (Lesson 09).

Zero, successor, natural number

So \(n\) follows \(m\) if some concept with number \(n\) loses one instance and then has number \(m\).

Proof sketch. Induction is Lesson 06. That \(0\) is not a successor: if \(S(0,n)\) with \(n=\#F\) and \(F(y)\), then \(\#F\) would equal the number of a concept with an instance, while \(0\) is the number of an empty concept; no bijection maps an empty concept onto a non-empty one, contradiction. Functionality and injectivity: a bijection between “\(F\) minus \(y\)” and “\(G\) minus \(z\)” extends by \(y\mapsto z\) to a bijection between \(F\) and \(G\); conversely, a bijection between \(F\) and \(G\) can be modified to send \(y\) to \(z\) and then restricted. The hard axiom is existence of successors. Frege’s idea (§§82–83) is to take the successor of \(n\) to be \(\#[x: S^{*=}(x,n)]\), the number of members of the number series ending with \(n\). One then proves by induction that no natural number follows itself, \(\neg S^*(n,n)\), and that \(S\big(n,\#[x:S^{*=}(x,n)]\big)\). The concept “number at most \(n\)” has one more instance than “number less than \(n\)”, whose number is \(n\).

George Boolos named the result in 1987. Charles Parsons had pointed out in 1965 that Hume’s principle suffices, and Crispin Wright gave a full reconstruction in 1983. Richard Heck later showed that in the Grundgesetze Frege himself derived the axioms essentially from Hume’s principle, using Basic Law V only to obtain it. Second-order logic plus Hume’s principle is consistent if second-order arithmetic is.

Concept, object, sense and reference

For the general reader. “The morning star is the evening star” records an astronomical discovery; “the morning star is the morning star” records nothing. Yet both names refer to Venus. Frege explained the difference by separating what an expression refers to from the way it presents its referent, its sense. The distinction became the starting point of the philosophy of language.

Function and concept (1891)

Frege extended the mathematical notion of function so that its values could be any objects. In particular, a concept is a function whose value is always a truth value, the True or the False, which he treated as objects. Thus \(x^2=1\) is a function of \(x\) with value the True for \(x=\pm1\) and the False otherwise. Functions are unsaturated: they have a gap for an argument. Objects are saturated. This is why, he argued, “the concept horse” refers to an object and not a concept, a paradox he accepted as an awkwardness of language.

Principle 2 forces principle 3 for sentences. If one replaces “the morning star” by “the evening star” in a sentence, the reference of the parts is unchanged, so the reference of the whole must be unchanged; what stays fixed under all such substitutions is the truth value. Principle 4 rescues compositionality in belief reports, where substituting co-referring names can change the truth value: inside “believes that”, the names refer to their senses, which differ.

Identity

In the Begriffsschrift, identity of content was a relation between names. In 1892 Frege rejected that view: \(a=b\) relates objects, and it is informative because \(a\) and \(b\) can present the same object in different ways. Names without a referent, like “Odysseus”, have sense and no reference. Sentences containing them then lack truth value, which Frege thought acceptable in fiction but a defect in a scientific language. Russell’s theory of descriptions (see the Russell course) was a direct response.

Basic Law V and Russell’s letter

For the general reader. To turn concepts into objects, Frege assumed that every concept has an extension, the class of things falling under it, and that two concepts have the same extension exactly when the same things fall under them. It looks harmless. In June 1902 Bertrand Russell showed that it is contradictory, using the class of all classes that are not members of themselves. Frege recognized the damage at once.

Frege defined membership by \(x\in y:\iff\exists G\,(y=\varepsilon G\wedge G(x))\). Basic Law V then implies naive comprehension: \(x\in\varepsilon F\iff F(x)\).

Proof. Let \(R(x)\) be \(\exists G\,(x=\varepsilon G\wedge\neg G(x))\), “\(x\) is the extension of a concept under which it does not fall”, and let \(r=\varepsilon R\). If \(R(r)\), there is \(G\) with \(r=\varepsilon G\) and \(\neg G(r)\). Then \(\varepsilon G=\varepsilon R\), so by Basic Law V \(G\) and \(R\) are coextensive, and \(\neg R(r)\). If \(\neg R(r)\), take \(G=R\): \(r=\varepsilon R\) and \(\neg R(r)\), so \(R(r)\). Either way, contradiction.

The comprehension principle of second-order logic, which provides the concept \(R\), is impredicative: \(R\) is defined by quantifying over all concepts, including \(R\) itself. Russell located the trouble there, and his theory of types (see the Russell course) restricts it. Frege located it in Basic Law V.

The letters

Russell’s letter of 16 June 1902, written in German, praised the Begriffsschrift and then set out the contradiction for a predicate that cannot be predicated of itself. Frege replied on 22 June. He called the discovery a great surprise and conceded that it shook the basis on which he had meant to build arithmetic. In an appendix to volume II (1903) he set out the paradox and proposed a weakened law, now called Frege’s Way Out:

It blocks Russell’s argument, but it is inconsistent with the existence of two distinct objects. Stanisław Leśniewski appears to have seen this in the 1930s, and W. V. Quine published a proof in 1955. Frege’s system has the two truth values, so the repair fails. There is no evidence that Frege found the flaw himself; he simply stopped developing the system.

Why Hume’s principle is different

Basic Law V maps concepts injectively into objects, which cannot be done when there are more concepts than objects (Cantor’s theorem). Hume’s principle maps only equinumerosity classes of concepts, and there are no more of these than cardinal numbers. In every infinite domain of size \(\kappa\), the cardinals up to \(\kappa\) can be represented by objects, so a model exists.

Axioms, geometry and Hilbert

For the general reader. In 1899 David Hilbert published axioms for geometry in which “point”, “line” and “between” could be read as anything that satisfies the axioms. Frege objected that axioms must be true statements about definite things, and that consistency does not create existence. Their exchange of letters is the first clear confrontation between the modern axiomatic method and the older view of axioms as self-evident truths.

The exchange

For Frege, a definition fixes the meaning of a new sign, and an axiom is a true thought whose truth is known independently. Hilbert’s axioms, he wrote in December 1899, were neither: they did not fix what a point is, so they could not be true or false. Hilbert answered that if axioms do not contradict one another, the things they define exist; and that any system of things satisfying them, “love, law, chimney-sweep”, would serve. Frege replied that the axioms then define a second-level concept, a condition on systems of relations, rather than first-level concepts. In the language of today, Hilbert’s axioms are a theory whose models form a class, which is exactly Frege’s diagnosis.

Frege published two series of articles “Über die Grundlagen der Geometrie” (1903 and 1906) in the Jahresbericht of the German Mathematical Society. In 1906 he sketched how one might prove the independence of a thought from others by a systematic translation of vocabulary that preserves logical form, an idea close to Hilbert’s method, though he doubted it could be made rigorous.

Sources of knowledge

In unpublished notes of 1924–1925, Frege abandoned logicism. He now held that arithmetic, like geometry, rests on a geometrical source of knowledge, and sketched a construction of numbers from the complex plane. The sketch was never developed.

The last years, 1903–1925

The second volume of the Grundgesetze appeared in 1903 with its appendix on the paradox. The planned third volume never came. Frege published polemics on the foundations of geometry and against formalist theories of real numbers, then fell largely silent. His wife Margarete died on 25 June 1904. The couple had lost children young. In 1908 he became guardian of Alfred and Toni Fuchs, aged five and three, and later adopted Alfred, who became an engineer.

He kept teaching. Rudolf Carnap took his courses on the concept-script and on logic in mathematics between 1910 and 1914 and later carried his ideas into logical empiricism. In 1911 the young Ludwig Wittgenstein visited him in Jena; Frege advised him to study with Russell in Cambridge. They met and corresponded again, and in 1919 Frege answered the manuscript of the Tractatus with puzzled objections. Frege retired in 1918 (some sources give 1917) and moved to Bad Kleinen, near Wismar. He published three essays under the title Logische Untersuchungen, beginning with “Der Gedanke” (1918–19), on thoughts as the timeless, mind-independent senses of sentences.

The diary

Between 10 March and 9 May 1924, Frege kept a political diary. It expresses monarchist, anti-democratic and nationalist views, hostility to socialists, Catholics and the French, and admiration for Erich Ludendorff. It also contains antisemitic passages: Frege wrote that Jews should be deprived of political rights and, ideally, removed from Germany. He agreed with an article that praised Hitler. These views were widespread on the German nationalist right in the 1920s, which explains them without excusing them. Michael Dummett, who read the diary among Frege’s papers, recorded his shock in the preface to his 1973 book on Frege. The text was published in German in the Deutsche Zeitschrift für Philosophie in 1994 and in English translation in Inquiry in 1996. The German Society for Analytic Philosophy debated renaming its Frege Prize in 2024–25; its members voted in 2025 to keep the name.

Frege died at Bad Kleinen on 26 July 1925, aged 76, and was buried in Wismar. He left his papers to Alfred, with a note that they contained things of lasting value. Most of the originals were destroyed in an Allied bombing raid on Münster in 1945; some survive in transcripts made before the war.

Rediscovery and legacy

During his life Frege’s main readers were Peano, Russell, Husserl, Wittgenstein and Carnap. Russell gave an appendix of The Principles of Mathematics (1903) to “the logical and arithmetical doctrines of Frege”, and through Russell and Whitehead’s Principia Mathematica his logic became the logic of the twentieth century, though often without his name. Wittgenstein acknowledged “the great works of Frege” in the preface to the Tractatus. Alonzo Church adopted the sense–reference distinction as a programme for intensional logic.

After the Second World War, translations by Peter Geach and Max Black (1952) and J. L. Austin’s translation of the Grundlagen (1950) made Frege a standard author in English-speaking philosophy. Dummett’s Frege: Philosophy of Language (1973) presented him as the founder of analytic philosophy. The neo-logicist programme of Crispin Wright and Bob Hale, after Boolos’s work on Frege’s theorem, revived the claim that arithmetic can be derived from Hume’s principle as an analytic truth.

Legacy

• Logic: the quantifier with variable binding and scope; the first formal system of higher-order logic; the idea of a gap-free proof.
• Foundations: the logicist programme; the ancestral and the logical definition of induction; Hume’s principle and Frege’s theorem.
• Philosophy of language: function and argument as the analysis of sentences; compositionality; sense and reference; the context principle.
• Computing: functions as first-class objects and the treatment of many-place functions as iterated one-place functions, both anticipating Church’s λ-calculus.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Gottlob Frege, MacTutor History of Mathematics.
2. Gottlob Frege, Wikipedia.
3. E. N. Zalta, Gottlob Frege, Stanford Encyclopedia of Philosophy.
4. Gottlob-Frege-Zentrum, Hochschule Wismar, Gottlob Frege (biographical leaflet).
5. L. Kreiser, Gottlob Frege: Leben, Werk, Zeit, Meiner, 2001.
6. G. Frege, List of Frege’s scientific correspondence, after the Wissenschaftlicher Briefwechsel (1976).
7. Gesellschaft für Analytische Philosophie, The Frege Prize in light of the antisemitic statements of its namesake.

Logic and mathematics

1. G. Frege, Begriffsschrift, Nebert, Halle, 1879. English translation in J. van Heijenoort (ed.), From Frege to Gödel, Harvard, 1967.
2. G. Frege, Die Grundlagen der Arithmetik, Koebner, Breslau, 1884; trans. J. L. Austin, 1950.
3. G. Frege, Über Sinn und Bedeutung, Zeitschrift für Philosophie und philosophische Kritik 100 (1892) 25–50.
4. G. Frege, Grundgesetze der Arithmetik, 2 vols., Pohle, Jena, 1893 and 1903; trans. P. Ebert and M. Rossberg, Oxford, 2013.
5. B. Russell, Letter to Frege (1902), and Frege’s reply, in van Heijenoort, From Frege to Gödel.
6. G. Boolos, The consistency of Frege’s Foundations of Arithmetic, in J. J. Thomson (ed.), On Being and Saying, MIT Press, 1987.
7. R. G. Heck, Frege’s Theorem, Oxford, 2011.
8. M. Dummett, Frege: Philosophy of Language, Duckworth, 1973; Frege: Philosophy of Mathematics, Duckworth, 1991.
9. W. V. Quine, On Frege’s way out, Mind 64 (1955) 145–159.
10. E. N. Zalta, Frege’s theorem and foundations for arithmetic, Stanford Encyclopedia of Philosophy.