Gottlob Frege

Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who became a logician and philosopher. He helped found both modern mathematical logic and analytic philosophy.

Life

Frege's father was a schoolteacher whose specialty was mathematics. Frege began his studies at the University of Jena in 1869, moving to Göttingen after two years, where he received his Ph.D. in mathematics, in 1873. According to Sluga (1980), the nature of Frege's university education in logic and philosophy is still unclear. In 1875, he returned to Jena as a lecturer. In 1879, he was made associate professor, and in 1896, professor. Frege had but one student of note, Rudolf Carnap. His children all having died before reaching maturity, he adopted a son in 1905.

Frege's work was not widely appreciated during his life, but the admiration of Bertrand Russell and Ludwig Wittgenstein, as well as Carnap, nonetheless guaranteed him significant influence in certain circles. His work became widely known in the English-speaking world only after World War II, in part because of the emigration to the United States of philosophers and logicians—Carnap, Alfred Tarski, and Kurt Gödel, for example—who knew and respected Frege's work and the appearance of translations into English of his major writings. Frege's work has since had enormous influence on analytic philosophy.

Logician

Frege is widely regarded as a logician on par with Aristotle, Kurt Gödel, and Alfred Tarski. His revolutionary Begriffsschrift, or Concept Script (1879) marked the beginning of a new epoch in the history of logic. The Begriffsschrift broke much new ground, including a clean treatment of functions and variables. Frege attempted to show mathematics as an extension of Aristotelian logic. He invented and axiomatized predicate logic, thanks to his discovery of quantified variables, which subsequently became ubiquitous in mathematics and solved the medieval problem of multiple generality. Hence the logical machinery essential to Bertrand Russell's theory of descriptions and Principia Mathematica (with Alfred North Whitehead), to Gödel's famous proof of the incompleteness theorem, was ultimately due to Frege.

Frege was a major advocate of the view that arithmetic is reducible to logic, a view known as logicism. In his Grundgesetze der Arithmetik (1893, 1903), published at its author's expense, he attempted to explicitly derive the laws of arithmetic from what he took to be logical axioms. Most of these were taken over from his Begriffsschrift, though that had undergone significant changes, as well. The one really new principle was Frege's Basic Law V, which said that the 'value-range' of a function f(x) is the same as the 'value-range' of the function g(x) if, and only if ∀x(fx = gx). As the second volume was about to go to press, Frege learned from Bertrand Russell that Russell's paradox could be derived from Basic Law V. Hence, the formal system of Grundgesetze was inconsistent. Frege gave a derivation of the contradiction in a last-minute appendix to volume two and attempted to remedy his system by modifying his Basic Law V, which was responsible for the contradiction. Frege's remedy to Basic Law V has been shown to be inconsistent (or, more precisely, to imply that there is only one object).

Recent work has shown that much of Frege's work can nonetheless be salvaged, in several different ways.

Basic Law V can be weakened in various ways that restore the consistency of the system. The best-known of these is due to George Boolos. Say that a 'concept' F is "small" if the objects falling under F cannot be put in 1-1 correspondence with the universe, that is, if: ¬∃Ris one-one & ∀x∃y(xRy & Fy). Now replaces Law V with the weaker claim, "New V", that 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). New V can be shown to be consistent if second-order arithmetic is and sufficient to allow proofs of the axioms of second-order arithmetic.
Replace Basic Law V 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 in one-one correspondence with the Gs. Again, this principle can be shown to be consistent if second-order arithmetic is and sufficient to allow proofs of the axioms of second-order arithmetic]. This result is known as Frege's Theorem.
The logic Frege uses, second-order logic, can be weakened to so-called predicative second-order logic. Such theories can be shown to be consistent by finitistic or constructive reasoning. Only very weak fragments of arithmetic can be interpreted in such systems, however.

Frege's work in logic was little recognized in his own day, in considerable part because his peculiar diagrammatic notation had no antecedents; it has since had no imitators. His ideas spread chiefly through those he influenced, particularly Bertrand Russell and Rudolf Carnap.

Philosopher

Frege is regarded as one of the founding fathers of analytic philosophy, mainly because of his conceptual contributions to the philosophy of language, such as his:

Function-argument analysis of the proposition;
Distinction between the sense and reference (Sinn und Bedeutung) of a proper name (Eigenname);
Advocacy of a mediated reference theory;
Distinction between concept and object (Begriff und Gegenstand);
Advancement of the context principle.
Formulation of the principle of compositionality.

As a philosopher of mathematics, Frege loathed appeals to psychologistic or "mental" explanations for meanings (such as idea theories of meaning). His original purpose was very far from answering questions about meaning; he wanted to use modern logic to further develop the foundations of arithmetic. He first undertook to answer the question "What is a number?" or "What objects do number-words ("one", "two", etc.) refer to?" But in pursuing these matters, he eventually faced the task of analysing and explaining what meaning is, and came to several major conclusions.

Frege, despite Bertrand Russell's generous praise, was little known as a philosopher during his lifetime. Here too, his ideas spread chiefly through those he influenced, including Ludwig Wittgenstein and Rudolf Carnap. He also studied, corresponded, and debated Edmund Husserl's works in print. Among some of the debates they had, Frege persuaded Husserl to abandon psychologism while Husserl criticized the way Frege used some of Leibniz's philosophy [http://perso.wanadoo.fr/rancho.pancho/Sub.htm.

References

Primary

Online bibliography of Frege's works and their English translations.
1879. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: Louis Nebert. Translation: Concept Script, a formal language of pure thought modelled upon that of arithmetic, by S. Bauer-Mengelberg in Jean Van Heijenoort, ed., 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press.
1884. Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. Translation: J. L. Austin, 1974. The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number, 2nd ed. Blackwell.
1891. "Funktion und Begriff." Translation: "Function and Concept" in Geach and Black (1980).
1892a. "Über Sinn und Bedeutung" in Zeitschrift für Philosophie und philosophische Kritik 100: 25-50. Translation: "On Sense and Reference" in Geach and Black (1980).
1892b. "Über Begriff und Gegenstand" in Vierteljahresschrift für wissenschaftliche Philosophie 16: 192-205. Translation: "Concept and Object" in Geach and Black (1980).
1893. Grundgesetze der Arithmetik, Band I. Jena: Verlag Hermann Pohle. Band II, 1903. Partial translation: Furth, M, 1964. The Basic Laws of Arithmetic. Uni. of California Press.
1904. "Was ist eine Funktion?" in Meyer, S., ed., 1904. Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20. Februar 1904. Leipzig: Barth: 656-666. Translation: "What is a Function?" in Geach and Black (1980).
Peter Geach and Max Black, eds., and trans., 1980. Translations from the Philosophical Writings of Gottlob Frege, 3rd ed. Blackwell.

Frege intended that the following three papers be published together in a book titled Logical Investigations. The English translations thereof were so published in 1975.

1918-19. "Der Gedanke: Eine logische Untersuchung (Thought: A Logical Investigation)" 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.

Secondary

George Boolos, 1998. Logic, Logic, and Logic. MIT Press. Contains several influential papers on Frege's philosophy of arithmetic and logic.
Michael Dummett, 1973. Frege: Philosophy of Language. Harvard University Press.
Michael Dummett, 1991. Frege: Philosophy of Mathematics. Harvard University Press.
Demopoulos, William, 1995. "Frege's Philosophy of Mathematics". Harvard University Press. A nice collection that explores the significance of Frege's theorem, and his mathematical and intellectural background.
Gillies, Douglas A., 1982. Frege, Dedekind, and Peano on the foundations of arithmetic. Assen, Netherlands: Van Gorcum.
Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press. Fair to the mathematician, less so to the philosopher.
Hatcher, William, 1982. The Logical Foundations of Mathematics. Pergamon. Uses natural deduction to rederive Peano's axioms from the Grundgesetze system, recast in modern notation.
Hill, C. O., and Rosado Haddock, G. E., 2000. Husserl or Frege: Meaning, Objectivity, and Mathematics. Open Court. The Frege-Husserl-Cantor triangle.
Hans Sluga, 1980. Gottlob Frege. Routledge.

External links