Background

According to the curriculum vitae that the 26-year old Frege filed in 1874 with his Habilitationsschrift, he was born on November 8, 1848, in Wismar, a town then in Mecklenburg-Schwerin but now in Mecklenburg-Vorpommern, Germany. His father, Alexander, a headmaster of a secondary school for girls, and his mother, Auguste (nee Bialloblotzky), brought him up in the Lutheran faith.

Education

Frege attended Große Stadtschule Wismar (now Geschwister Scholl Gymnasium Wismar) for 15 years, and after graduation in 1869, entered the University of Jena. At Jena, Frege attended lectures by Ernst Karl Abbe, who subsequently became Frege's mentor and who had a significant intellectual and personal influence on Frege's life.

Frege transferred to the University of Göttingen in 1871, and two years later, in 1873, was awarded a doctoral degree in mathematics, having written a dissertation under Ernst Schering titled Über eine geometrische Darstellung der imaginären Gebilde in der Ebene ("On a Geometrical Representation of Imaginary Forms in the Plane"). Frege explains the project in his thesis as follows: "By a geometrical representation of imaginary forms in the plane we understand accordingly a kind of correlation in virtue of which every real or imaginary element of the plane has a real, intuitive element corresponding to it." Here, by "imaginary forms," Frege is referring to imaginary points, imaginary curves, and lines, etc. Interestingly, one section of the thesis concerns the representation of complex numbers by magnitudes of angles in the plane.

Career

In 1874, Frege completed his Habilitationsschrift, entitled Rechnungsmethoden, die sich auf eine Erweiterung des Grössenbegriffes gründen ("Methods of Calculation Based on an Extension of the Concept of Quantity"). Immediately after submitting this thesis, the good offices of Abbe led Frege to become a Privatdozent (Lecturer) at the University of Jena. Library records from the University of Jena establish that, over the next 5 years, Frege checked out texts in mechanics, analysis, geometry, Abelian functions, and elliptical functions. No doubt, many of these texts helped him to prepare the lectures he is listed as giving by the University of Jena course bulletin, for these lectures are on topics that often match the texts, i.e., analytic geometry, elliptical and Abelian functions, algebraic analysis, functions of complex variables, etc.

In 1879, Frege published his first book Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Concept Notation: A formula language of pure thought, modeled upon that of arithmetic) and was promoted to außerordentlicher Professor (Extraordinarius Professor) at Jena. Although the Begriffsschrift constituted a major advance in logic, it was neither widely understood nor well-received. Some scholars have suggested that this was due to the facts that the notation was 2-dimensional instead of linear and that he didn't build upon the work of others but rather presented something radically new.

Frege's next really significant work was his second book, Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriff der Zahl, published in 1884. Frege begins this work with criticisms of previous attempts to define the concept of number, and then offers his own analysis. The Grundlagen contains a variety of insights still discussed today, such as: (a) the claim that a statement of number is a higher-order assertion about a concept; (b) his famous Context Principle ("never ask for the meaning of a word in isolation, but only in the context of a proposition"), and (c) the formulation of a principle (now called "Hume's Principle" in the secondary literature) that asserts the equivalence of the claim "the number of Fs is equal to the number of Gs" with the claim that "there is a one-to-one correspondence between the objects falling under F and the objects falling under G." More generally, Frege provides in the Grundlagen a non-technical philosophical justification and outline of the ideas that he was to develop technically in his two-volume work Grundgesetze der Arithmetik.

In the years 1891-1892, Frege published three of his most well-known papers, Function and Concept, On Sense and Reference, and On Concept and Object. Immediately after that, in 1893, he published the first volume of the technical work previously mentioned, Grundgesetze der Arithmetik. In 1896, he was promoted to ordentlicher Honorarprofessor (regular honorary professor). Six years later (on June 16, 1902), as he was preparing the proofs of the second volume of the Grundgesetze, he received a letter from Bertrand Russell, informing him that one could derive a contradiction in the system he had developed in the first volume. Russell's letter frames the paradox first in terms of the predicate P = "being a predicate which cannot be predicated of itself," and then in terms of the class of all those classes that are not members of themselves. Frege, in the Appendix to the second volume, rephrased the paradox in terms of his own system.

Frege never fully recovered from the fatal flaw discovered in the foundations of his Grundgesetze. His attempts at salvaging the work by restricting Basic Law V were not successful. However, he continued teaching at Jena, and from 1903-1917, he published six papers, including What is a Function? (1904) and On the Foundations of Geometry. In the latter, Frege criticized Hilbert's understanding and use of the axiomatic method. In 1917, he retired from the University of Jena.

In the last phase of Frege's life, from 1917 to 1925, Frege published three philosophical papers, in a series, with the titles The Thought, Negation, and Compound Thoughts. After that, however, we have only fragments of philosophical works. Unfortunately, his last years saw him become more than just politically conservative and right-wing - his diary for a brief period in 1924 show sympathies for fascism and anti-Semitism. He died on July 26, 1925, in Bad Kleinen (now in Mecklenburg-Vorpommern).

Achievements

• Gottlob Frege is widely known as a mathematician and logician, who founded modern mathematical logic. Working on the borderline between philosophy and mathematics Frege discovered, on his own, the fundamental ideas that have made possible the whole modern development of logic and thereby invented an entire discipline.
  
  Frege's work was not widely appreciated during his lifetime, although he did debate in print, and correspond with, Ernst Schroder, Peano, Husserl, and Cantor. Bertrand Russell discovered Frege's writings around 1900 and became an admirer, as did Ludwig Wittgenstein somewhat later. These admirers assured Frege's influence and reputation in certain restricted circles. Frege had but one student of note, albeit a consequential one: Rudolf Carnap. Frege's work became widely known in the English-speaking world only after World War II; English translations of Frege's writings, which first appeared in 1950, came to have an enormous influence on analytic philosophy. Frege also became better-known thanks to the emigration to the United States of central European philosophers and logicians who knew and respected Frege's work, including Carnap, Alfred Tarski, and Kurt Gödel.

Works

More photos

• book

  • The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number

    (The Foundations of Arithmetic is undoubtedly the best int...)

    1893
  • Collected Papers on Mathematics, Logic, and Philosophy

    (Widely published on logic, analysis, geometry, and arithm...)

    1991
  • Gottlob Frege: Foundations of Arithmetic 1884

Religion

Gottlob Frege was brought up in the Lutheran faith.

Politics

Gottlob Frege was right-wing in his political views, and like many conservatives of his generation in Germany, he is known to have been distrustful of foreigners and rather anti-semitic. Frege seems to have wanted to see all Jews expelled from Germany, or at least deprived of certain political rights. This distasteful feature of Frege's personality has gravely disappointed some of Frege's intellectual progeny.

Views

Frege is widely regarded today as a logician on a par with Aristotle, Kurt Gödel, and Alfred Tarski. His 1879 Begriffsschrift (Concept Script) marked a turning point in the history of logic. The Begriffsschrift broke much new ground, including a clean treatment of functions and variables. Frege wanted to show that mathematics grew out of Aristotelian logic, but in so doing devised techniques that took him far beyond that logic. In effect, he invented axiomatic predicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and solved the medieval problem of multiple generality in which traditional logic could not prove certain intuitively obvious inferences. Hence the logical machinery essential to Bertrand Russell's theory of descriptions and Principia Mathematica (with Alfred North Whitehead), and to Gödel's incompleteness theorems, is 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 derive 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 f(x) is the same as the "value-range" of the function g(x) if and only if ∀x[f(x) = g(x)].

As Vol. 2 of the Grundgesetze was about to go to press in 1903, Bertrand Russell wrote to Frege, showing how to derive [Russell's paradox] from Basic Law V. (This letter and Frege's reply thereto are translated in Jean van Heijenoort 1967.) Russell had shown that the system of the Grundgesetze was 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's proposed remedy was subsequently shown to imply that there is but one object in the universe of discourse, and hence is worthless.

Recent work has shown, however, that much of the Grundgesetze can be salvaged in any of several ways:

(1) Basic Law V can be weakened in ways that restore the consistency of Frege's system. The best-known way is due to George Boolos. A "concept" F is classed as "small" if the objects falling under F cannot be put in 1-to-1 correspondence with the universe of discourse, that is, if: ¬∃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.

(2) 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 into a one-to-one correspondence with the Gs. This principle too is consistent for second-order arithmetic and suffices to prove the axioms of second-order arithmetic. This result is anachronistically termed Frege's Theorem.

(3) Frege's logic, now known as second-order logic, can be weakened to so-called predicative second-order logic. However, this logic, although provably consistent by finitistic or constructive methods, can interpret only very weak fragments of arithmetic.

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; 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 analyzing 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 employed as a mathematician, not a philosopher, and 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 and the first collections of his writings appeared only after WWII. Hence despite Bertrand Russell's generous praise, Frege was little known as a philosopher during his lifetime. His ideas spread chiefly through those he influenced, such as Russell, Wittgenstein, and Rudolf Carnap, and through Polish work on logic and semantics.

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

"A scientist can hardly encounter anything more desirable than, just as a work is completed, to have its foundation give way."

"[...] I do not begin with concepts and put them together to form a thought or judgement; I come by the parts of a thought by analysing the thought."

Personality

Although Frege was a fierce, sometimes even satirical, polemicist, he himself was a quiet, reserved man.

Gottlob Frege's students remember him as someone who spent much time on the chalkboard rather than interacting with the pupils, however, it is said that he was slightly bitter and sarcastic during his lectures.

Interests

• Reading

• Philosophers & Thinkers

  Bertrand Russell, Ludwig Wittgenstein

Connections

In 1887, Frege married Margarete Katharina Sophia Anna Lieseberg. The couple had no children. There was an adopted son, Alfred, however, who became an engineer.

Father:

Alexander Frege

Mother:

Auguste Wilhelmine Sophie Frege

Spouse:

Margarete Katharina Sophia Anna Lieseberg

(married 1887)

adopted son:

Alfred

Doctoral advisor:

Ernst Christian Julius Schering

Student:

Rudolf Carnap

Rudolf Carnap was a German-born American philosopher of logical positivism. He made important contributions to logic, the analysis of language, the theory of probability, and the philosophy of science.

References

• Gottlob Frege: Leben - Werk - Zeit by Lothar Kreiser
  2001