Logicism is an attitude towards mathematics according to which it is an extension of logic and therefore all mathematical concepts and theories are reducible to logic. If this program were feasible, it could support logical positivism in particular, and reductionism in general. Bertrand Russell and Alfred North Whitehead defended this approach, created by mathematician Gottlob Frege.
Logicism played a key role in the development of analytical philosophy in the twentieth century.
Frege, Russell and Whitehead
Louis Couturat, Bertrand Russell and Alfred North Whitehead supported this theory created by Gottlob Frege. Gottlob Frege abandoned the project after Russell discovered a paradox brought to light by a contradiction in naive set theory. Russell and Whitehead continued the project in their book Principia Mathematica.
Although arithmetic was reduced by Cantor to set theory, set theory itself could never be derived from pure logic. However, Gödel’s incompleteness theorem, discovered in 1931, showed that any system rich enough to formalize arithmetic would contain truths which could not be demonstrated within this system. This put an end to the initial program of logicism.
Although the ambition of this reductionist project thus had to be revised downwards, the major part of modern mathematics continues today to be thought by many mathematicians and logicians as being reducible to a logic which would be based on axioms of Zermelo-Fraenkel theory, which does not present, for the moment, any known contradictions. There is thus a neo-logicism, which is based in particular on the so-called “Hume principle”, and defended in particular by Crispin Wright and Bob Hale.