Gottlob Frege | Biography

Gottlob Frege made seminal contributions to the philosophy of language and of mathematics, yet his major intellectual project ran aground as it was in the last stages of being launched. He worked in a fairly narrow area of mathematics throughout his career, although his contributions to philosophy were uncommonly acute for a mathematician. Frege's understanding of language continues to be the focus for philosophical discussion and his project in mathematical logic, though unsuccessful, provided the foundation for much further work in the subject.

Friedrich Ludwig Gottlob Frege was born on November 8, 1848, in Wismar, Mecklenburg-Scherin (present-day Germany). His father, Alexander, was principal of a girls' high school. In 1869 Frege enrolled at the University of Jena, where he stayed for two years at Jena before transferring to the University of Göttingen. There, his studies were not limited to mathematics, but included physics, chemistry, and philosophy. In 1873 Frege earned his doctorate from Göttingen; his thesis examined the question of the geometric representation of imaginary structures in the plane. The choice of subject was in keeping with the geometrical aspect of German mathematical research at the time, one year after the inauguration of the Erlangen program by Felix Klein. A year later, Frege earned the right to teach at the University of Jena with a dissertation dealing with the subject of groups of a certain kind of function. Frege's work on group theory continued, but he never made it the focus of his courses as did some of his contemporaries.

Reconstructs the Logical World

The goal to which Frege devoted himself was the task of reducing arithmetic to logic. There were two main currents in mathematics which led him to find the project attractive and feasible. The first was the work of Karl Weierstrass and his school of analysis. Weierstrass had taken the work of two centuries in the area of calculus and determined a way to rewrite the foundations arithmetically. The other current was the mathematization of logic in the work of George Boole and other algebraists. After centuries of little progress, logic had begun to take on some of the qualities of modern mathematics.

As a first step, Frege felt it necessary to devise a new notation in order to prevent hidden assumptions from creeping into arguments, which was published as Begriffsschrift("Concept Notation"), in 1879. This notation was a tool for analyzing and representing proofs within mathematics, but it was cumbersome and typographically awkward. As a result, Frege's notation did not find many supporters, although the project was taken up by others, including Bertrand Russell, with more success.

The next step for Frege was to prove that the current philosophy of mathematics was flawed and in need of revamping, an argument he detailed in his 1884 book Grundlagen der Arithmetik("The Foundations of Arithemetic"). Frege took on psychologism, the doctrine that numbers were merely objects in the mind, and was able to refute the claims of its current advocates. Similarly, he assailed the empiricism of John Stuart Mill, which sought to find numbers as objects in the world of sense and sound. Both psychologism and empiricism survived Frege's onslaught, but he had clearly demonstrated some of their shortcomings. The Grundlagen also included Frege's definition of number. It is tied to the notion of concept and deals with the question of the existence of a 1-1 correspondence between objects falling under two concepts.

Frege found it necessary to address a few philosophical issues connected with concepts, objects, and how they are represented linguistically. His essays dealing with these subjects have a continuing philosophical appeal independent of the success of his logical project. In particular, the 1892 essay, "über Sinn und Bedeutung" ("On Sense and Reference") may be the most influential paper in the history of the philosophy of language.

The first volume of the Grundgesetze der Arithmetik("The Basic Laws of Arithmetic") was published in 1893 and it appeared that Frege had finally constructed a system that reduced arithmetic to logic. In 1902, however, he received a letter from Bertrand Russell, who had detected an error in Frege's system. What Russell had found was a contradiction, a derivation of both a statement and its negation. According to the rules of Frege's logic, the presence of such a contradiction rendered the system useless.

Frege's response was intellectually honest. In the second volume of the Grundgesetze, Frege acknowledged Russell's contribution and noted that his own system was unable to handle it. For the remainder of his career, Frege worked on issues that were connected with philosophy and logic without seeking to repair his system. He retired from the University of Jena in 1917 and died at the German resort of Bad Kleinen on July 26, 1925. The subsequent interest in Frege's work, especially among philosophical logicians, bears witness to the value of his reformulation of logic and the farsightedness of his ambition.