Mr. Friedrich Ludwig Gottlob Frege

A German mathematician who became a logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. As a philosopher, he is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and mathematics. While he was mainly ignored by the intellectual world when he published his writings, Giuseppe Peano (1858?1932) and Bertrand Russell (1872?1970) introduced his work to later generations of logicians and philosophers. Though his education and early work were mathematical, especially geometrical, his thought soon turned to logic. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Halle a/S: Verlag von Louis Nebert, 1879) (Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic) marked a turning point in the history of logic. The Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables. He wanted to show that mathematics grows out of logic, but in so doing he devised techniques that took him far beyond the Aristotelian syllogistic and Stoic propositional logic that had come down to him in the logical tradition. In effect, he invented axiomatic predicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality. Previous logic had dealt with the logical constants and, or, if ... then ..., not, and some and all, but iterations of these operations, especially "some" and "all", were little understood. It is frequently noted that Aristotle's logic is unable to represent even the most elementary inferences in Euclid's geometry, but his "conceptual notation" can represent inferences involving indefinitely complex mathematical statements. The analysis of logical concepts and the machinery of formalization that is essential to Principia Mathematica (3 vols., 1910?1913) (by Bertrand Russell, 1872?1970, and Alfred North Whitehead, 1861?1947), to Russell's theory of descriptions, to Kurt G?del's (1906?1978) incompleteness theorems, and to Alfred Tarski's (1901?1983) theory of truth, is ultimately due to him. One of his stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to "intuition". If there was an intuitive element, it was to be isolated and represented separately as an axiom: from there on, the proof was to be purely logical and without gaps. Having exhibited this possibility, his larger purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism: unlike geometry, arithmetic was to be shown to have no basis in "intuition", and no need for non-logical axioms. Already in the 1879 Begriffsschrift important preliminary theorems, for example a generalized form of law of trichotomy, were derived within what he understood to be pure logic. His work in logic had little international attention until 1903 when Russell wrote an appendix to The Principles of Mathematics stating his differences with him. The diagrammatic notation that he used had no antecedents (and has had no imitators since). Moreover, until Russell and Whitehead's Principia Mathematica (3 vols.) appeared in 1910?13, the dominant approach to mathematical logic was still that of George Boole (1815?1864) and his intellectual descendants, especially Ernst Schr?der (1841?1902). His logical ideas nevertheless spread through the writings of his student Rudolf Carnap (1891?1970) and other admirers, particularly Bertrand Russell and Ludwig Wittgenstein (1889?1951).