Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.

Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano 1858–1932, Bertrand Russell 1872–1970, and, to some extent, Ludwig Wittgenstein 1889–1951 introduced his work to later generations of philosophers.

His contributions include the development of modern logic in the Begriffsschrift and work in the foundations of mathematics. His book the Foundations of Arithmetic is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical papers "On Sense and Reference" and "The Thought" are also widely cited. The former argues for two different types of meaning and descriptivism. In Foundations and "The Thought", Frege argues for Platonism against psychologism or formalism, concerning numbers and propositions respectively. Russell's paradox undermined the logicist project by showing Frege's Basic Law V in the Foundations to be false.

## Life

### Childhood 1848–69

Frege was born in 1848 in Wismar, Mecklenburg-Schwerin today part of Mecklenburg-Vorpommern. His father Carl Karl Alexander Frege 1809–1866 was the co-founder and headmaster of a girls' high school until his death. After Carl's death, the school was led by Frege's mother Auguste Wilhelmine Sophie Frege née Bialloblotzky, 12 January 1815 – 14 October 1898; her mother was Auguste Amalia Maria Ballhorn, a descendant of Philipp Melanchthon and her father was Johann Heinrich Siegfried Bialloblotzky, a descendant of a Polish noble family who left Poland in the 17th century.

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, entitled Hülfsbuch zum Unterrichte in der deutschen Sprache für Kinder von 9 bis 13 Jahren 2nd ed., Wismar 1850; 3rd ed., Wismar and Ludwigslust: Hinstorff, 1862 Help book for teaching German to children from 9 to 13 years old, the first section of which dealt with the structure and logic of language.

Frege studied at Große Stadtschule Wismar and graduated in 1869. His teacher Gustav Adolf Leo Sachse 5 November 1843 – 1 September 1909, who was 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 1869–74

Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Confederation. In the four semesters of his studies he attended approximately twenty courses of lectures, most of them on mathematics and physics. His most important teacher was Ernst Karl Abbe 1840–1905; physicist, mathematician, and inventor. Abbe gave lectures on theory of gravity, galvanism and electrodynamics, complex analysis 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 Carl Zeiss AG, 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 Christian Philipp Karl Snell 1806–86; subjects: use of infinitesimal analysis in geometry, analytic geometry of planes, analytical mechanics, optics, physical foundations of mechanics; Hermann Karl Julius Traugott Schaeffer 1824–1900; analytic geometry, applied physics, algebraic analysis, on the telegraph and other electronic machines; and the philosopher Kuno Fischer 1824–1907; 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 Rudolf Friedrich Alfred Clebsch 1833–72; analytic geometry, Ernst Christian Julius Schering 1824–97; function theory, Wilhelm Eduard Weber 1804–91; physical studies, applied physics, Eduard Riecke 1845–1915; theory of electricity, and Hermann Lotze 1817–81; 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 Christian Julius Schering, with a dissertation under the title of "Ueber 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.

Frege married Margarete Katharina Sophia Anna Lieseberg 15 February 1856 – 25 June 1904 on 14 March 1887.

## Work as a logician

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 , 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.

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. 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 fx is the same as the "value-range" of the function gx if and only if ∀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. 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∃yxRy & 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 ∀xFx ↔ 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".
- 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.

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.

## Philosopher

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.

## Sense and reference

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".

## 1924 diary

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." 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. 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. 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, who greatly valued his teaching, and it was he who encouraged Ludwig Wittgenstein to leave for England in order to study with Bertrand Russell. The 1924 diary was published posthumously in 1994. Frege apparently never spoke in public about his political viewpoints.

## Personality

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.

## Important dates

- 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.

## Important works

### Logic, foundation of arithmetic

Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens 1879, Halle an der Saale: Verlag von Louis Nebert online version.

- In English: Begriffsschrift, a Formula Language, Modeled Upon That of Arithmetic, for Pure Thought, in: J. van Heijenoort ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Harvard, MA: Harvard University Press, 1967, pp. 5–82.
- In English selected sections revised in modern formal notation: R. L. Mendelsohn, The Philosophy of Gottlob Frege, Cambridge: Cambridge University Press, 2005: "Appendix A. Begriffsschrift in Modern Notation: 1 to 51" and "Appendix B. Begriffsschrift in Modern Notation: 52 to 68."

Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl 1884, Breslau: Verlag von Wilhelm Koebner online version.

- In English: The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, translated by J. L. Austin, Oxford: Basil Blackwell, 1950.

Grundgesetze der Arithmetik, Band I 1893; Band II 1903, Jena: Verlag Hermann Pohle online version.

- In English translation of selected sections, "Translation of Part of Frege's Grundgesetze der Arithmetik," translated and edited Peter Geach and Max Black in Translations from the Philosophical Writings of Gottlob Frege, New York, NY: Philosophical Library, 1952, pp. 137–158.
- In German revised in modern formal notation: Grundgesetze der Arithmetik, Korpora portal of the University of Duisburg-Essen, 2006: Band I and Band II.
- In German revised in modern formal notation: Grundgesetze der Arithmetik – Begriffsschriftlich abgeleitet. Band I und II: In moderne Formelnotation transkribiert und mit einem ausführlichen Sachregister versehen, edited by T. Müller, B. Schröder, and R. Stuhlmann-Laeisz, Paderborn: mentis, 2009.
- In English: Basic Laws of Arithmetic, translated and edited with an introduction by Philip A. Ebert and Marcus Rossberg. Oxford: Oxford University Press, 2013. ISBN 978-0-19-928174-9.

### Philosophical studies

"Function and Concept" 1891

- Original: "Funktion und Begriff", an address to the Jenaische Gesellschaft für Medizin und Naturwissenschaft, Jena, 9 January 1891.
- In English: "Function and Concept".

"On Sense and Reference" 1892

- Original: "Über Sinn und Bedeutung", in Zeitschrift für Philosophie und philosophische Kritik C 1892: 25–50.
- In English: "On Sense and Reference", alternatively translated in later edition as "On Sense and Meaning".

"Concept and Object" 1892

- Original: "Ueber Begriff und Gegenstand", in Vierteljahresschrift für wissenschaftliche Philosophie XVI 1892: 192–205.
- In English: "Concept and Object".

"What is a Function?" 1904

- Original: "Was ist eine Funktion?", in Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20 February 1904, S. Meyer ed., Leipzig, 1904, pp. 656–666.
- In English: "What is a Function?".

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, the papers were published together in Logische Untersuchungen, ed. G. Patzig, Vandenhoeck & Ruprecht, 1966, and English translations appeared together in Logical Investigations, ed. Peter Geach, Blackwell, 1975.

- 1918–19. "Der Gedanke: Eine logische Untersuchung" "The Thought: A Logical Inquiry", in Beiträge zur Philosophie des Deutschen Idealismus I: 58–77.
- 1918–19. "Die Verneinung" "Negation" in Beiträge zur Philosophie des Deutschen Idealismus I: 143–157.
- 1923. "Gedankengefüge" "Compound Thought", in Beiträge zur Philosophie des Deutschen Idealismus III: 36–51.

### Articles on geometry

- 1903: "Über die Grundlagen der Geometrie". II. Jahresbericht der deutschen Mathematiker-Vereinigung XII 1903, 368–375. In English: "On the Foundations of Geometry".
- In English: "On the Foundations of Geometry".
- 1967: Kleine Schriften. I. Angelelli, ed.. Darmstadt: Wissenschaftliche Buchgesellschaft, 1967 and Hildesheim, G. Olms, 1967. "Small Writings," a collection of most of his writings e.g., the previous, posthumously published.

The Foundations of Arithmetic (1884)

Basic Laws of Arithmetic (1893–1903)