Frege , (Friedrich Ludwig) Gottlob

(1848–1925) German philosopher and mathematician

Born at Wismar (now in Germany), Frege studied at the universities of Jena and Göttingen, where he obtained his PhD in 1873. He then returned to Jena as a lecturer, where he remained for the rest of his working life, rising to the position of professor in 1896. In a series of seminal works Frege laid the foundations of modern mathematical logic, transforming logic with an understanding of and notation for the problem of multiple generality – propositions containing predicates, quantifiers, and variables – and showing how the basic concepts and operations of mathematics could be formalized. He also revolutionized modern philosophy through his influence on the philosophy of language. However Frege's work was almost completely ignored, misunderstood, or treated with hostility by his contemporaries – notable exceptions were Bertrand Russell and Giuseppe Peano.

In his first major work, Begriffsschrift (Concept Writing, 1879), he provided a new formalism containing an adequate symbolism and an axiomatic base for the rigorous derivation of both propositional and predicate logic. While a few workers, such as the American C.S. Pierce, had been moving in this direction, their work was completely overshadowed by the comprehensive nature of Frege's work.

In Die Grundlagen der Arithmetik (1884; The Foundations of Arithmetic) Frege gave a formal definition of cardinal number and showed how basic properties of numbers could be logically derived from it. In Grundgesetze der Arithmetik (1893 and 1903; Basic Laws of Arithmetic) he went further in attempting to derive arithmetic from formal logic. The Grundgesetze is still regarded as a massive achievement, but his main aim was doomed to failure. On the eve of the publication of the second volume Russell wrote to Frege pointing out a contradiction – Russell's paradox – that could be derived from his system. This, as Frege acknowledged, vitiated his whole project.