Introduction
Logicism was advocated by Richard Dedekind, developed by Gottlob Frege and extended by Bertrand Russell (1872–1970) together with Alfred North Whitehead (1861–1947). We shall not say anything about Dedekind here, but take up the philosophical position starting with Frege. Frege developed logicism through three works. The first, the Begriffsschrift (Concept Script) (1976), first published in 1879, is a technical work, introducing the reader to a formal logical system. The second work, the Grundlagen (Foundations of Arithmetic) (1980a), first published in 1884, is philosophical. The third, the Grundgesetze der Arithmetik (1980b), was originally published in two volumes, in 1893 and 1903. It would also be translated as Foundations of Arithmetic, but these are formal foundations, not philosophical ones. Whitehead and Russell continued the logicist project, and published Principia Mathematica (1910–13) in three volumes. This is a technical work developing a formal theory of types, which, they argue, is pure logic. Russell also published more philosophical works: The Principles of Mathematics (1903) and Introduction to Mathematical Philosophy (1919).
The major philosophical question the logicist tries to answer is: what is the essence of mathematics? As the word “logicist” suggests, the answer is that mathematics, or part of it, is essentially logic. Logicism can be either a realist philosophy of mathematics or an anti-realist philosophy of mathematics. Frege was a realist. Frege's logicist believes that mathematical truths are independent of human beings. Frege's version of logicism is epistemologically realist. That is, human beings discover, or fail to discover, mathematical truths.