(Student Guide to World Philosophy)

In 1879, Gottlob Frege, a German mathematician and logician, published a slim volume entitled Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (“Begriffsschrift, a Formula Language, Modeled upon That of Arithmetic, for Pure Thought,” 1967; better known as Conceptual Notation, 1972), in which, for the first time in the history of logic, the fundamental ideas and principles of mathematical logic were set out. In this remarkable book, Frege achieved a much deeper analysis of deductive inference than had previous logicians; for example, the problem of deductive inferences involving multiply embedded expressions of generality (for example, “everyone loves someone”) was finally solved. Frege realized that the formal system of logic he had devised was precisely the instrument needed for realizing philosopher Gottfried Wilhelm Leibniz’s dream of reducing arithmetic to logic. Frege believed that before attempting this daunting task, an informal overview and defense of the project should be given; The Foundations of Arithmetic was written for that purpose. Frege later attempted the rigorous reduction of the concepts and truths of arithmetic to those of logic in Grundgesetze der Arithmetik, 1893-1903 (two volumes; partial translation, The Basic Laws of Arithmetic, 1964).