Gottlob Frege

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Lore & Background

Frege was born in 1848 in Wismar, Mecklenburg-Schwerin. His father, Carl Alexander Frege, 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. Frege studied at Große Stadtschule Wismar and graduated in 1869. Teacher of mathematics and natural science Gustav Adolf Leo Sachse encouraged him to continue his studies at the University of Jena. Frege matriculated at the University of Jena in the spring of 1869. His most important teacher was Ernst Karl Abbe, who became a trusted friend. Starting in 1871, Frege continued his studies in Göttingen, where he attended lectures of Alfred Clebsch, Ernst Christian Julius Schering, Wilhelm Eduard Weber, Eduard Riecke, and Hermann Lotze. In 1873, Frege attained his doctorate under Schering. Frege married Margarete Lieseberg on 14 March 1887. The couple had at least two children, who died when young. Years later, they adopted a son, Alfred. His 1879 book Begriffsschrift marked a turning point in the history of logic, breaking new ground with a rigorous treatment of the ideas of functions and variables. Frege invented axiomatic predicate logic, in large part thanks to his invention of quantified variables, which solved the problem of multiple generality. His goal was to show that mathematics grows out of logic, a view known as logicism. This idea was formulated in non-symbolic terms in his The Foundations of Arithmetic (1884). Later, in his Basic Laws of Arithmetic (vol. 1, 1893; vol. 2, 1903), Frege attempted to derive, by use of his symbolism, all of the laws of arithmetic from axioms he asserted as logical. The one truly new principle was 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)]. In a famous episode, Bertrand Russell wrote to Frege, just as Vol. 2 of the Grundgesetze was about to go to press in 1903, showing that Russell's paradox could be derived from Frege's Basic Law V. Frege wrote a hasty, last-minute Appendix to Vol. 2, deriving the contradiction and proposing to eliminate it by modifying Basic Law V.

Reader's Guide

Frege's significance lies in his foundational contributions to modern logic and the philosophy of mathematics. His Begriffsschrift introduced axiomatic predicate logic and quantified variables, which became essential to later works such as Principia Mathematica, Russell's theory of descriptions, Gödel's incompleteness theorems, and Tarski's theory of truth. His book The Foundations of Arithmetic is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical papers 'On Sense and Reference' and 'The Thought' are also widely cited. The former argues for two different types of meaning and descriptivism. In Foundations and 'The Thought', Frege argues for Platonism against psychologism or formalism, concerning numbers and propositions respectively. Though largely ignored during his lifetime, his work was introduced to later generations by Peano, Russell, and Wittgenstein, and he is now widely considered one of the greatest logicians since Aristotle and one of the most profound philosophers of mathematics ever.

Did You Know?

• Frege's father, Carl Alexander Frege, co-founded and was headmaster of a girls' high school, and wrote a textbook on the German language for children aged 9 to 13.
• Frege's mother, Auguste Wilhelmine Sophie Frege, led the school after her husband's death; her mother was a descendant of Philipp Melanchthon.
• Frege's most important teacher at Jena was Ernst Karl Abbe, who later became a trusted friend and, as director of Carl Zeiss AG, was in a position to advance Frege's career.
• Frege attained his doctorate in 1873 under Ernst Christian Julius Schering at the University of Göttingen.
• Bertrand Russell wrote to Frege in 1903, just as Vol. 2 of the Grundgesetze was about to go to press, showing that Russell's paradox could be derived from Frege's Basic Law V.