Frege, Gottlob

(1848–1925)

German mathematician and philosopher of mathematics. Frege was born in the small town of Wismar in Pomerania, and was sent to the university of Jena when he was twenty-one. He obtained his doctorate at Göttingen, and worked almost the whole of his life in the mathematics department at the university of Jena. His first important work, the Begriffsschrift (‘concept writing’ 1879), is also the first example of a formal system in the sense of modern logic. In it Frege undertakes to devise a formal system within which mathematical proofs may be given. It was his discovery of the correct representation of generality, the notion of quantifier and variable, that opened the possibility of successfully achieving this aim. With that notation Frege could represent sentences involving multiple generality (such as the form ‘for every small number e there is a number n such that…’) on which the validity of much mathematical reasoning depends. The Begriffsschrift also contains the elements of the propositional calculus, including an informal presentation of the notion of a truth-function . It is universally acknowledged to mark the beginning of modern logic. In 1884 Frege published the Grundlagen der Arithmetik (trs. as The Foundations of Arithmetic by J. L. Austin, 1959). In this work, Frege brilliantly attacks rival accounts of the status of arithmetic, and then propounds his own approach to the subject, analysing the basic concepts of mathematics in such a form that a reduction of arithmetic to operations that are fundamentally logical in nature becomes a real possibility. The first volume of the Grundgesetze der Arithmetik (1893, trs. as The Basic Laws of Arithmetic, 1964) formalizes the mathematical approach of the Grundlagen, a task that necessitated giving the first formal theory of classes; it was this theory that was later shown inconsistent by Russell's paradox . Volume ii of the Grundgesetze, concerned mainly with the theory of real numbers, was published in 1903. Frege's own reaction to Russell's paradox, after understandable initial consternation, was to modify one of his own axioms; the result, however, was not eventually tenable, and it was only with Zermelo's work that the modern conception of set theory was put on a satisfactory footing.

Frege's distinction as a logician is matched by his deep concern with the basic semantic concepts involved in the logical foundations of his work. In a succession of papers he forges the basic concepts and distinctions that have dominated subsequent philosophical investigation of logic and language. The topics of these writings include sense (Sinn ) and reference, concepts, functions and objects, identity, negation, assertion, truth/falsity, and the nature of thought. Although his relations to the philosophical surroundings of his time are debatable, these concerns and his approach to them stamp Frege as the founding figure of analytical philosophy . However, his concern to protect a timeless objectivity for thought and its contents has led to accusations of Platonism, and his own views of the objects of mathematics troubled him until the end of his life. Translations include Translations from the Philosophical Writings of Gottlob Frege, edited by P. Geach and M. Black (1960), The Basic Laws of Arithmetic, translated and edited by M. Furth (1964), Conceptual Notation and Related Articles, edited by T. W. Bynum (1972), and On the Foundations of Geometry and Formal Theories of Arithmetic, edited by E.-H. W. Kluge (1971).