Logicism in the context of Philosophy of mathematics

Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician, and public intellectual. He influenced mathematics, logic, set theory, and various areas of analytic philosophy.

He was one of the early 20th century's prominent logicians and a founder of analytic philosophy, along with his predecessor Gottlob Frege, his friend and colleague G. E. Moore, and his student and protégé Ludwig Wittgenstein. Russell with Moore led the British "revolt against idealism". Together with his former teacher Alfred North Whitehead, Russell wrote Principia Mathematica, a milestone in the development of classical logic and a major attempt to reduce the whole of mathematics to logic (see logicism). Russell's article "On Denoting" has been considered a "paradigm of philosophy".

Julius Wilhelm Richard Dedekind (/ˈdeɪdɪkɪnd/; German: [ˈdeːdəˌkɪnt]; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as logicism.

As the study of argument is of clear importance to the reasons that we hold things to be true, logic is of essential importance to rationality. Arguments may be logical if they are "conducted or assessed according to strict principles of validity", while they are rational according to the broader requirement that they are based on reason and knowledge.

Logic and rationality have each been taken as fundamental concepts in philosophy. They are not the same thing. Philosophical rationalism in its most extreme form is the doctrine that knowledge can ultimately be founded on pure reason, while logicism is the doctrine that mathematical concepts, among others, are reducible to pure logic.

The Foundations of Arithmetic (German: Die Grundlagen der Arithmetik) is a book by Gottlob Frege, published in 1884, which investigates the philosophical foundations of arithmetic. Frege refutes other idealist and materialist theories of number and develops his own platonist theory of numbers. The Grundlagen also helped to motivate Frege's later works in logicism.

The book was also seminal in the philosophy of language. Michael Dummett traces the linguistic turn to Frege's Grundlagen and his context principle.

Ludwig Wittgenstein considered his chief contribution to be in the philosophy of mathematics, a topic to which he devoted much of his work between 1929 and 1944. As with his philosophy of language, Wittgenstein's views on mathematics evolved from the period of the Tractatus Logico-Philosophicus, as he changed from logicism (which was endorsed by his mentor Bertrand Russell) towards a general anti-foundationalism and constructivism that was not readily accepted by the mathematical community. The success of Wittgenstein's general philosophy has tended to displace the debates on more technical issues.

His Remarks on the Foundations of Mathematics contains his compiled views, notably a controversial repudiation of Gödel's incompleteness theorems.