The German mathematician and philosopher Gottlob Frege (1848-1925) is considered the founder of modern mathematical logic. His work was almost wholly ignored during his lifetime but now exerts a great influence on the philosophy of logic and language.

Gottlob Frege was born on Nov. 8, 1848, at Wismar. He began his university studies at Jena in 1869 but after 2 years moved to Göttingen. He studied mathematics, the natural sciences, and philosophy and took his degree in 1873. Thereafter he taught at Jena in the department of mathematics. He was made a professor in 1896 and retired in 1918. Frege was married to Margarete Lieseberg, and the couple had one adopted son. Frege died on July 26, 1925, in Bad Kleinen.

Frege invented the concept of a formal system of mathematical logic, and in his first major work, *Begriffsschrift* (1879), he presented the first example of such a system in his formulation of a propositional and predicate calculus. He introduced the mathematical notion of function and variable into logic and invented the idea of quantifiers. He was
also the first writer on axiomatic theory to make clear the distinction between an axiom and a rule of inference.

Further progress in this work convinced Frege that the basic ideas of arithmetic (but not of geometry) could be articulated solely in logical expressions. He expressed his new program first in a nonsymbolic work, *The Foundations of Arithmetic* (1884), which also featured a brilliant and devastating polemic against all previous attempts at the subject. The crown of his work was to be his *Basic Laws of Arithmetic.* The first volume of this work appeared in 1893; but in 1903, as Frege was about to issue the second volume, Bertrand Russell pointed out a contradiction in Frege's use of the concept of a "class," which undermined the proofs in the work. Frege hastily added an appendix that sought to remedy the defect (this effort was later proved defective), but thereafter he seemed to lose interest in the great project. Two decades later he regarded the whole enterprise as an error and fell back upon the Kantian interpretation of mathematical judgments as synthetic a priori.

Frege also made important contributions to the philosophy of logic. Concerning the old question: what is it for a proposition to have meaning?—he introduced a variety of distinctions that are being exploited by contemporary philosophers. Frege rejected epistemology as the starting point of philosophy and revived the classical view, dominant before René Descartes, that philosophical logic holds this place.

## Further Reading on Gottlob Frege

There are no biographies of Frege. His work, however, has been extensively studied, especially since translations of it have become available, beginning in the 1950s. A convenient collection of most of the important critical essays is in E. D. Klemke, ed., *Essays on Frege* (1968), which also has a complete bibliography. Two difficult but rewarding full-length studies are Jeremy D. B. Walker, *A Study of Frege* (1965), and Robert Sternfeld, *Frege's Logical Theory* (1966).