1.1 Childhood (1848–69) 1.2 Studies at University: Jena and Göttingen (1869–74)
2 Work as a logician 3 Philosopher 4 Sense and reference 5 1924 diary 6 Personality 7 Important dates 8 Important works
8.1 Logic, foundation of arithmetic 8.2 Philosophical studies 8.3 Articles on geometry
9 See also 10 References
10.1 Primary 10.2 Secondary
11 External links
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
Title page to
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
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". 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
Function–argument analysis of the proposition; Distinction between concept and object (Begriff und Gegenstand); Principle of compositionality; Context principle; 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
It should be kept in mind that Frege was employed as 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
Main article: 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
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 contains
extreme right-wing 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 and Adolf Hitler. 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
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
Habilitation at Jena; private teacher.
1879 — Ausserordentlicher Professor at Jena.
1896 — Ordentlicher Honorarprofessor at Jena.
1917 or 1918 — retires.
Died 26 July 1925 in
Important works Logic, foundation of arithmetic Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (1879). Halle a. S.
Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl (1884). Breslau.
English: The Foundations of Arithmetic: A Logical-Mathematical
Investigation of the
Grundgesetze der Arithmetik, Band I (1893); Band II (1903). Jena: Verlag Hermann Pohle.
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"; in 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".
Original: "Ueber Begriff und Gegenstand", in Vierteljahresschrift für
wissenschaftliche Philosophie XVI (1892): 192–205;
In English: "
"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 (Internet Archive: , , ); 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".
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.
Frege programming language List of pioneers in computer science
^ Hans Sluga, "Frege's alleged realism," Inquiry 20 (1–4):227–242
^ a b Michael Resnik, "II. Frege as Idealist and then Realist,"
Inquiry 22 (1–4):350–357 (1979).
^ Truth – Internet Encyclopedia of Philosophy; The Deflationary
Theory of Truth (Stanford Encyclopedia of Philosophy).
^ Willard Van Orman Quine, introduction to "Bausteine der
mathematischen Logik", pp. 305–316. Translated by Stefan
Bauer-Mengelberg as "On the building blocks of mathematical logic" in
Jean van Heijenoort
Online bibliography of Frege's works and their English translations
(compiled by E. N. Zalta, Stanford Encyclopedia of Philosophy).
1879. Begriffsschrift, eine der arithmetischen nachgebildete
Formelsprache des reinen Denkens. Halle a. S.: Louis Nebert.
Badiou, Alain. "On a Contemporary Usage of Frege", trans. Justin
Clemens and Sam Gillespie. UMBR(a), no. 1, 2000, pp. 99–115.
Baker, Gordon, and P.M.S. Hacker, 1984. Frege: Logical Excavations.
Oxford University Press. — Vigorous, if controversial,
criticism of both Frege's philosophy and influential contemporary
interpretations such as Dummett's.
Currie, Gregory, 1982. Frege: An Introduction to His Philosophy.
Dummett, Michael, 1973. Frege: Philosophy of Language. Harvard
------, 1981. The Interpretation of Frege's Philosophy. Harvard
Hill, Claire Ortiz, 1991. Word and Object in Husserl, Frege and
Russell: The Roots of Twentieth-Century Philosophy. Athens OH: Ohio
------, and Rosado Haddock, G. E., 2000. Husserl or Frege: Meaning,
Objectivity, and Mathematics. Open Court. — On the
Kenny, Anthony, 1995. Frege — An introduction to the founder of
modern analytic philosophy. Penguin Books. — Excellent
non-technical introduction and overview of Frege's philosophy.
Klemke, E.D., ed., 1968. Essays on Frege. University of Illinois
Press. — 31 essays by philosophers, grouped under three
headings: 1. Ontology; 2. Semantics; and 3.
Anderson, D. J., and Edward Zalta, 2004, "Frege, Boolos, and Logical
Objects," Journal of Philosophical
Everdell, William R. (1997), The First Moderns: Profiles in the Origins of Twentieth Century Thought, Chicago: University of Chicago Press
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